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Wheelbase — Wikipedia Republished // WIKI 2
Distance between the centers of the front and rear wheels
For the British TV series of the same name, see Wheelbase (TV series).
Bike geometry parameters: The wheelbase of a bicycle
In both road and rail vehicles, the wheelbase is the horizontal distance between the centers of the front and rear wheels. For road vehicles with more than two axles (e.g. some trucks), the wheelbase is the distance between the steering (front) axle and the centerpoint of the driving axle group. In the case of a tri-axle truck, the wheelbase would be the distance between the steering axle and a point midway between the two rear axles.[1]
Very Fast Mechanical Mini Car vs Simplest Electromagnetic Train
Rail inspection car with 4 stroke engine
Thomas Tank Engine, Brio, Subway Tunnel, Rail Crossing, Wooden Toy Vehicles for Kids, Children Songs
1.1 Varying wheelbases within nameplate
1.2 Bikes
1.3 Skateboards
2 Rail
The wheelbase of a vehicle equals the distance between its front and rear wheels. At equilibrium, the total torque of the forces acting on a vehicle is zero. Therefore, the wheelbase is related to the force on each pair of tires by the following formula:
{\displaystyle F_{f}={d_{r} \over L}mg}
{\displaystyle F_{r}={d_{f} \over L}mg}
{\displaystyle F_{f}}
is the force on the front tires,
{\displaystyle F_{r}}
is the force on the rear tires,
{\displaystyle L}
is the wheelbase,
{\displaystyle d_{r}}
is the distance from the center of mass (CM) to the rear wheels,
{\displaystyle d_{f}}
is the distance from the center of mass to the front wheels (
{\displaystyle d_{f}}
{\displaystyle d_{r}}
{\displaystyle L}
{\displaystyle m}
is the mass of the vehicle, and
{\displaystyle g}
is the gravity constant. So, for example, when a truck is loaded, its center of gravity shifts rearward and the force on the rear tires increases. The vehicle will then ride lower. The amount the vehicle sinks will depend on counter acting forces, like the size of the tires, tire pressure, and the spring rate of the suspension. If the vehicle is accelerating or decelerating, extra torque is placed on the rear or front tire respectively. The equation relating the wheelbase, height above the ground of the CM, and the force on each pair of tires becomes:
{\displaystyle F_{f}={d_{r} \over L}mg-{h_{cm} \over L}ma}
{\displaystyle F_{r}={d_{f} \over L}mg+{h_{cm} \over L}ma}
{\displaystyle F_{f}}
{\displaystyle F_{r}}
{\displaystyle d_{r}}
is the distance from the CM to the rear wheels,
{\displaystyle d_{f}}
is the distance from the CM to the front wheels,
{\displaystyle L}
{\displaystyle m}
is the mass of the vehicle,
{\displaystyle g}
is the acceleration of gravity (approx. 9.8 m/s2),
{\displaystyle h_{cm}}
is the height of the CM above the ground,
{\displaystyle a}
is the acceleration (or deceleration if the value is negative). So, as is common experience, when the vehicle accelerates, the rear usually sinks and the front rises depending on the suspension. Likewise, when braking the front noses down and the rear rises.:[2]
Because of the effect the wheelbase has on the weight distribution of the vehicle, wheelbase dimensions are crucial to the balance and steering. For example, a car with a much greater weight load on the rear tends to understeer due to the lack of the load (force) on the front tires and therefore the grip (friction) from them. This is why it is crucial, when towing a single-axle caravan, to distribute the caravan's weight so that down-thrust on the tow-hook is about 100 pounds force (400 N). Likewise, a car may oversteer or even "spin out" if there is too much force on the front tires and not enough on the rear tires. Also, when turning there is lateral torque placed upon the tires which imparts a turning force that depends upon the length of the tire distances from the CM. Thus, in a car with a short wheelbase ("SWB"), the short lever arm from the CM to the rear wheel will result in a greater lateral force on the rear tire which means greater acceleration and less time for the driver to adjust and prevent a spin out or worse.
Wheelbases provide the basis for one of the most common vehicle size class systems.
Varying wheelbases within nameplate
Some luxury vehicles are offered with long-wheelbase variants to increase the spaciousness and therefore the luxury of the vehicle. This practice can often be found on full-size cars like the Mercedes-Benz S-Class, but ultra-luxury vehicles such as the Rolls-Royce Phantom and even large family cars like the Rover 75 came with 'limousine' versions. Prime Minister of the United Kingdom Tony Blair was given a long-wheelbase version of the Rover 75 for official use.[3] and even some SUVs like the VW Tiguan and Jeep Wrangler come in LWB models
In contrast, coupé varieties of some vehicles such as the Honda Accord are usually built on shorter wheelbases than the sedans they are derived from.
Main article: Bicycle and motorcycle geometry
The wheelbase on many commercially available bicycles and motorcycles is so short, relative to the height of their centers of mass, that they are able to perform stoppies and wheelies.
In skateboarding the word 'wheelbase' is used for the distance between the two inner pairs of mounting holes on the deck. This is different from the distance between the rotational centers of the two wheel pairs. A reason for this alternative use is that decks are sold with prefabricated holes, but usually without trucks and wheels. It is therefore easier to use the prefabricated holes for measuring and describing this characteristic of the deck.
A common misconception is that the choice of wheelbase is influenced by the height of the skateboarder. However, the length of the deck would then be a better candidate, because the wheelbase affects characteristics useful in different speeds or terrains regardless of the height of the skateboarder. For example, a deck with a long wheelbase, say 22 inches (55.9 cm), will respond slowly to turns, which is often desirable in high speeds. A deck with a short wheelbase, say 14 inches (35.6 cm), will respond quickly to turns, which is often desirable when skating backyard pools or other terrains requiring quick or intense turns.
In rail vehicles, the wheelbase follows a similar concept. However, since the wheels may be of different sizes (for example, on a steam locomotive), the measurement is taken between the points where the wheels contact the rail, and not between the centers of the wheels.
On vehicles where the wheelsets (axles) are mounted inside the vehicle frame (mostly in steam locomotives), the wheelbase is the distance between the front-most and rear-most wheelsets.
On vehicles where the wheelsets are mounted on bogies (American: trucks), three wheelbase measurements can be distinguished:
the distance between the pivot points of the front-most and rear-most bogie;
the distance between the front-most and rear-most wheelsets of the vehicle;
the distance between the front-most and rear-most wheelsets of each bogie.
The wheelbase affects the rail vehicle's capability to negotiate curves. Short-wheelbased vehicles can negotiate sharper curves. On some larger wheelbase locomotives, inner wheels may lack flanges in order to pass curves.
The wheelbase also affects the load the vehicle poses to the track, track infrastructure and bridges. All other conditions being equal, a shorter wheelbase vehicle represents a more concentrated load to the track than a longer wheelbase vehicle. As railway lines are designed to take a pre-determined maximum load per unit of length (tonnes per meter, or pounds per foot), the rail vehicles' wheelbase is designed according to their intended gross weight. The higher the gross weight, the longer the wheelbase must be.
Bicycle and motorcycle geometry
Static equilibrium
Wheel gauge
^ ISO 8855:2011, 4.2
^ Ruina, Andy; Rudra Pratap (2002). Introduction to Statics and Dynamics (PDF). Oxford University Press. p. 350. Retrieved 2007-03-23.
^ Biggs, Henry (October 27, 2004). "Rover 75". autoexpress.co.uk. Auto Express. Retrieved May 8, 2017.
wheelbase (P3039) (see uses)
|
1UCLA Department of Mathematics, Marina Del Rey, USA.
2University of Hawaii, Honolulu, USA.
Counting has always been one of the most important operations for human be-ings. Naturally, it is inherent in economics and business. We count with the unique arithmetic, which humans have used for millennia. However, over time, the most inquisitive thinkers have questioned the validity of standard arithmetic in certain settings. It started in ancient Greece with the famous philosopher Zeno of Elea, who elaborated a number of paradoxes questioning popular knowledge. Millennia later, the famous German researcher Herman Helmholtz (1821-1894) [1] expressed reservations about applicability of conventional arithmetic with respect to physical phenomena. In the 20th and 21st century, mathematicians such as Yesenin-Volpin (1960) [2], Van Bendegem (1994) [3], Rosinger (2008) [4] and others articulated similar concerns. In validation, in the 20th century expressions such as 1 + 1 = 3 or 1 + 1 = 1 occurred to reflect important characteristics of economic, business, and social processes. We call these expressions synergy arithmetic. It is common notion that synergy arithmetic has no meaning mathematically. However in this paper we mathematically ground and explicate synergy arithmetic.
Synergy, Synergy Arithmetic, Non-Diophantine Arithmetics, Mergers and Acquisitions
Burgin, M. and Meissner, G. (2017) 1 + 1 = 3: Synergy Arithmetic in Economics. Applied Mathematics, 8, 133-144. doi: 10.4236/am.2017.82011.
N=\left\{1,2,3,\cdots \right\}
a\oplus b=g\left(f\left(a\right)+f\left(b\right)\right)
a\circ b=g\left(f\left(a\right)\cdot f\left(b\right)\right)
\oplus
\circ
A=〈A,\oplus ,\circ 〉
\oplus
\circ
\oplus
\circ
g\circ f
g\left(x\right)=⌈x⌉
f\left(x\right)=2x
g\left(x\right)=3x
{h}_{\text{T}}\left(x\right)=⌈h\left(x\right)⌉
{h}^{\text{T}}\left(x\right)=⌊{h}^{-1}\left(x\right)⌋
⌈a⌉
⌊a⌋
⌈2.75⌉=3
⌊2.75⌋=2
f={h}_{\text{T}}
g={h}^{\text{T}}
AW=〈W,\oplus ,\circ 〉
a\oplus b={h}^{\text{T}}\left({h}_{\text{T}}\left(a\right)+{h}_{\text{T}}\left(b\right)\right)
a\circ b={h}^{\text{T}}\left({h}_{\text{T}}\left(a\right)\cdot {h}_{\text{T}}\left(b\right)\right)
{h}_{\text{T}}\left({0}_{\mu }\right)=0
h\left(x\right)
a\le b
{h}_{\text{T}}\left(Sa\right)-{h}_{\text{T}}\left(a\right)\le {h}_{\text{T}}\left(Sb\right)-{h}_{\text{T}}\left(b\right)
f\left(x\right)=2x
{f}^{-1}\left(x\right)=\left(1/2\right)x
{f}_{\text{T}}\left(a\right)=f\left(a\right)
{f}^{\text{T}}\left(c\right)={f}^{-1}\left(c\right)
A=〈W,\oplus ,\circ 〉
1\oplus 1=\left(1/2\right)\left(2\cdot 1+2\cdot 1\right)=\left(1/2\right)\left(2+2\right)=\left(1/2\right)\left(4\right)=2
2\oplus 2=\left(1/2\right)\left(2\cdot 2+2\cdot 2\right)=\left(1/2\right)\left(4+4\right)=\left(1/2\right)\left(8\right)=4
1\circ 1=\left(1/2\right)\left(\left(2\cdot 1\right)\cdot \left(2\cdot 1\right)\right)=\left(1/2\right)\left(2\cdot 2\right)=\left(1/2\right)\left(4\right)=2
f\left(x\right)=x+5
{f}^{-1}\left(x\right)=x-5
2\circ 2=\left(1/2\right)\left(\left(2\cdot 2\right)\cdot \left(2\cdot 2\right)\right)=\left(1/2\right)\left(4\cdot 4\right)=\left(1/2\right)\left(16\right)=8
f\left(x\right)=x+1
{f}^{-1}\left(x\right)=x-1
{f}_{\text{T}}\left(a\right)=f\left(a\right)
{f}^{\text{T}}\left(c\right)={f}^{-1}\left(c\right)
A=〈W,\oplus ,\circ 〉
1\oplus 1=\left(\left(1+1\right)+\left(1+1\right)\right)-1=\left(2+2\right)-1=4-1=3
2\oplus 2=\left(\left(2+1\right)+\left(2+1\right)\right)-1=\left(3+3\right)-1=6-1=5
1\circ 1=\left(\left(1+1\right)\cdot \left(1+1\right)\right)-1=\left(2\cdot 2\right)-1=4-1=3
2\circ 2=\left(\left(2+1\right)\cdot \left(2+1\right)\right)-1=\left(3\cdot 3\right)-1=9-1=8
f\left(x\right)={\mathrm{log}}_{2}x
{f}^{-1}\left(x\right)={2}^{x}
A=〈W,\oplus ,\circ 〉
1\oplus 1={2}^{\left({\mathrm{log}}_{2}1+{\mathrm{log}}_{2}1\right)}={2}^{\left(0+0\right)}={2}^{0}=1
2\oplus 2={2}^{\left({\mathrm{log}}_{2}2+{\mathrm{log}}_{2}2\right)}={2}^{\left(1+1\right)}={2}^{2}=4
1\circ 1={2}^{\left({\mathrm{log}}_{2}1\cdot {\mathrm{log}}_{2}1\right)}={2}^{\left(0\cdot 0\right)}={2}^{0}=1
2\circ 2={2}^{\left({\mathrm{log}}_{2}2\cdot {\mathrm{log}}_{2}2\right)}={2}^{\left(1\cdot 1\right)}={2}^{1}=2
f\left(x\right)=x-1
{f}^{-1}\left(x\right)=x+1
{f}_{\text{T}}\left(a\right)=f\left(a\right)
{f}^{\text{T}}\left(c\right)={f}^{-1}\left(c\right)
A=〈W,\oplus ,\circ 〉
1\oplus 1=\left(\left(1-1\right)+\left(1-1\right)\right)+1=\left(0+0\right)+1=0+1=1
2\oplus 2=\left(\left(2-1\right)+\left(2-1\right)\right)+1=\left(1+1\right)+1=2+1=3
2\oplus 1=\left(\left(2-1\right)+\left(1-1\right)\right)+1=\left(1+0\right)+1=1+1=2
1\circ 1=\left(\left(1-1\right)\cdot \left(1-1\right)\right)+1=\left(0\cdot 0\right)+1=0+1=1
2\circ 2=\left(\left(2-1\right)\cdot \left(2-1\right)\right)+1=\left(1\cdot 1\right)+1=1+1=2
a\ll b
b\gg a
a\ll b
b\oplus a=b
b\oplus a=b
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Formal and informal definitions.
Multi-tape turing machines.
Formal definition of multi-tape turing machines.
In the prerequisite articles, we learned about regular and context-free languages. We learned of computational devices that accept or generate these languages and their limitations when it comes to languages such as; A = {
{\mathrm{a}}^{\mathrm{m}}
{\mathrm{b}}^{\mathrm{n}}
{\mathrm{c}}^{\mathrm{mn}}
: m ≥ 0, n ≥ 0}.
A turing machine models a real computer. These machines accept all context-free grammar as well as languages such as A.
In this article, we will try to prove that "every problem solvable by a real computer is solvable by a turing machine" - this is the church turing thesis.
A turing machine will consist of the following;
First we have k tapes, each tape divided into cells. This tape is infinite on both sides left and right. Each cell in the tape stores a symbol that belongs to a finite set T referred to as the tape alphabet. This alphabet has a blank symbol ⋄ If a cell has this symbol, it is considered empty.
Let's look at an example of a turing machine where k is 2;
Each tape has a tape that can be moved along the other tape. It only moves a single cell per move. It reads the cell it is currently positioned at and replaces the symbol in the cell with another symbol.
There is also a state control that can be any of a finite number of states. A finite set of states is denoted by Q, this set has three special states namely, start, accept, and reject.
During a single step of computation the turing machine does the following;
At the beginning it is in a state r of Q and each k tape head is on a specific cell.
Depending on the current state r and k symbols;
The machine switches to state r' of Q
Each tape head writes a symbol T in the cell it is currently scanning.
Each tape head can move one cell to the left, right, or stay where it currently is.
A deterministic turing machine is 7-tuple consisting of M = (Σ, Γ, Q, δ, q,
{\mathrm{q}}_{\mathrm{accept}}
{\mathrm{q}}_{\mathrm{reject}}
Σ is a finite set referred to as an input alphabet.
Γ is a finite set referred to as a tape alphabet, it has the blank symbol ⋄ and Σ ⊆ Γ.
Q is a finite set whose elements are referred to as states.
q is an element of Q referred to as the start state.
{\mathrm{q}}_{\mathrm{accept}}
is an element of Q, it is referred to as the accept state.
{\mathrm{q}}_{\mathrm{reject}}
is also an element of Q, it is referred to as the reject state.
δ is referred to as a transition function.
δ:Q ×
{\mathrm{\Gamma }}^{\mathrm{k}}
→ Q ×
{\mathrm{\Gamma }}^{\mathrm{k}}
{\mathrm{\left\{L, R, N\right\}}}^{\mathrm{k}}
This function is a program of the turing machine, it tells the machine what it can do in one computation step.
These machines have multiple tapes each accessed with a separate head that moves independently despite the other heads.
In the beginning, the input is at tape 1 while others remain blank. In the next step, the machine reads consecutive symbols under its head and prints a symbol on each tape then moves its head.
Although it is easier to obtain a two-tape machine compared to a single-tape turing machine, this doesn't mean that a two-tape machine is more powerful.
A formal definition of multi-tape turing machines.
This machine is 6-tuple (Q, X, B, δ, q0, F) where;
δ is a relation between states and symbols where;
δ: Q × Xk → Q × (X × {Leftshift, Rightshift, Noshift })k where there is k number of tapes.
Note that for every multi-tape turing machine, there is an equivalent single-tape turing machine.
An algorithm is a list of steps describing how to solve a problem with a computer and therefore any computational process that can be specified by a program is considered an algorithm.
Similarly, a turing machine specifies a computational process and therefore we consider it as an algorithm.
Now, we ask. Is it possible to give a mathematical definition of an algorithm?
Having stated that every program be it a java program or C program represents an algorithm and that every turing machine represents an algorithm. We ask ourselves, are these two statements regarding an algorithm equivalent?
The answer is yes, it also implies that many different notions of computational processes are equivalent.
For example, the following are computational models.
One-tape turing machines.
Non-deterministic turing machines.
k-tape turing machines where k >= 1.
Java, C++ programs.
Any of the models can be converted to any other model.
The Church-Turing thesis: It states that every computational process that is intuitively considered an algorithm can be converted to a turing machine.
In other terms, we define an algorithm to be a turing machine.
A turing machine is a mathematical model of a computation that defines an abstract machine. Despite its simplicity, given any algorithm, this machine is capable of implementing the algorithm's logic.
The Church-Turing thesis states that every computational process that is said to be an algorithm can be implemented by a turing machine.
Variations of turing machines
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Biorthogonal wavelet filter set - MATLAB biorfilt - MathWorks 日本
biorfilt
Biorthogonal Filters and Transfer Functions
LoD1,HiD1,LoR1,HiR1
Biorthogonal Filters
Biorthogonal wavelet filter set
[LoD,HiD,LoR,HiR] = biorfilt(DF,RF)
[LoD1,HiD1,LoR1,HiR1,LoD2,HiD2,LoR2,HiR2] = biorfilt(DF,RF,'8')
[LoD,HiD,LoR,HiR] = biorfilt(DF,RF) returns four filters associated with the biorthogonal wavelet specified by decomposition filter DF and reconstruction filter RF. These filters are
[LoD1,HiD1,LoR1,HiR1,LoD2,HiD2,LoR2,HiR2] = biorfilt(DF,RF,'8') returns eight filters, the first four associated with the decomposition wavelet, and the last four associated with the reconstruction wavelet.
This example shows how to obtain the decomposition (analysis) and reconstruction (synthesis) filters for the 'bior3.5' wavelet.
Obtain the two scaling and wavelet filters associated with the 'bior3.5' wavelet.
wv = 'bior3.5';
[Rf,Df] = biorwavf(wv);
[LoD,HiD,LoR,HiR] = biorfilt(Df,Rf);
Plot the filter impulse responses.
title(['Dec. Lowpass Filter ',wv])
title(['Dec. Highpass Filter ',wv])
title(['Rec. Lowpass Filter ',wv])
title(['Rec. Highpass Filter ',wv])
Demonstrate that autocorrelations at even lags are only zero for dual pairs of filters. Examine the autocorrelation sequence for the lowpass decomposition filter.
npad = 2*length(LoD)-1;
LoDxcr = fftshift(ifft(abs(fft(LoD,npad)).^2));
lags = -floor(npad/2):floor(npad/2);
stem(lags,LoDxcr,'markerfacecolor',[0 0 1])
set(gca,'xtick',-10:2:10)
Examine the cross-correlation sequence for the lowpass decomposition and synthesis filters. Compare the result with the preceding figure. At even lags, the cross-correlation is zero.
xcr = fftshift(ifft(fft(LoD,npad).*conj(fft(LoR,npad))));
stem(lags,xcr,'markerfacecolor',[0 0 1])
title('Cross-correlation')
Compare the transfer functions of the analysis and synthesis scaling and wavelet filters.
dftLoD = fft(LoD,64);
dftLoD = dftLoD(1:length(dftLoD)/2+1);
dftHiD= fft(HiD,64);
dftHiD = dftHiD(1:length(dftHiD)/2+1);
dftLoR = fft(LoR,64);
dftLoR = dftLoR(1:length(dftLoR)/2+1);
dftHiR = fft(HiR,64);
dftHiR = dftHiR(1:length(dftHiR)/2+1);
df = (2*pi)/64;
freqvec = 0:df:pi;
plot(freqvec,abs(dftLoD),freqvec,abs(dftHiD),'r')
title('Transfer Modulus - Dec. Filters')
plot(freqvec,abs(dftLoR),freqvec,abs(dftHiR),'r')
title('Transfer Modulus - Rec. Filters')
DF — Decomposition scaling filter
Decomposition scaling filter associated with a biorthogonal wavelet, specified as a vector.
RF — Reconstruction scaling filter
Reconstruction scaling filter associated with a biorthogonal wavelet, specified as a vector.
LoD,HiD — Decomposition filters
Wavelet decomposition filters, returned as a pair of even-length real-valued vectors. LoD is the lowpass decomposition filter, and HiD is the highpass decomposition filter.
LoR,HiR — Reconstruction filters
Wavelet reconstruction filters, returned as a pair of even-length real-valued vectors. LoR is the lowpass reconstruction filter, and HiR is the highpass reconstruction filter.
LoD1,HiD1,LoR1,HiR1 — Filters
Filters associated with the decomposition (analysis) wavelet, returned as even-length real-valued vectors.
LoD1 — Decomposition lowpass filter
HiD1 — Decomposition highpass filter
LoR1 — Reconstruction lowpass filter
HiR1 — Reconstruction highpass filter
Filters associated with the reconstruction (synthesis) wavelet, returned as even-length real-valued vectors.
It is well known in the subband filtering community that if the same FIR filters are used for reconstruction and decomposition, then symmetry and exact reconstruction are incompatible (except with the Haar wavelet). Therefore, with biorthogonal filters, two wavelets are introduced instead of just one.
One wavelet,
\stackrel{Ë}{\mathrm{Ï}}
, is used in the analysis, and the coefficients of a signal s are
{\stackrel{Ë}{c}}_{j,k}=â«s\left(x\right){\stackrel{Ë}{\mathrm{Ï}}}_{j,k}\left(x\right)dx
The other wavelet, ψ, is used in the synthesis:
s=\underset{j,k}{â}{\stackrel{Ë}{c}}_{j,k}{\mathrm{Ï}}_{j,k}
Furthermore, the two wavelets are related by duality in the following sense:
â«{\stackrel{Ë}{\mathrm{Ï}}}_{j,k}\left(x\right){\mathrm{Ï}}_{{j}^{â²},{k}^{â²}}\left(x\right)dx=0
as soon as j ≠j′ or k ≠k′ and
â«{\stackrel{Ë}{\mathrm{Ï}}}_{0,k}\left(x\right){\mathrm{Ï}}_{0,{k}^{â²}}\left(x\right)dx=0
as soon as k ≠k′.
It becomes apparent, as A. Cohen pointed out in his thesis (p. 110), that “the useful properties for analysis (e.g., oscillations, null moments) can be concentrated in the
\stackrel{Ë}{\mathrm{Ï}}
function; whereas, the interesting properties for synthesis (regularity) are assigned to the ψ function. The separation of these two tasks proves very useful.â€
\stackrel{Ë}{\mathrm{Ï}}
and ψ can have very different regularity properties, ψ being more regular than
\stackrel{Ë}{\mathrm{Ï}}
\stackrel{Ë}{\mathrm{Ï}}
, ψ,
\stackrel{Ë}{\mathrm{Ï}}
and Ï• functions are zero outside a segment.
[1] Cohen, Albert. "Ondelettes, analyses multirésolution et traitement numérique du signal," Ph. D. Thesis, University of Paris IX, DAUPHINE. 1992.
biorwavf | orthfilt
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High School Calculus/Derivatives of Trigonometric Functions - Wikibooks, open books for an open world
High School Calculus/Derivatives of Trigonometric Functions
Formulas for Differentiation of Trigonometric Functions[edit | edit source]
In the following formulas the angle u is supposed to be expressed in circular measure.
{\displaystyle {\operatorname {d} \over \operatorname {d} x}\sin u=\cos u}
{\displaystyle {\operatorname {d} \over \operatorname {d} x}\cos u=-\sin u}
{\displaystyle {\operatorname {d} \over \operatorname {d} x}\tan u=\sec ^{2}u}
{\displaystyle {\operatorname {d} \over \operatorname {d} x}\cot u=-\csc ^{2}u}
{\displaystyle {\operatorname {d} \over \operatorname {d} x}\sec u=\sec u\tan u}
{\displaystyle {\operatorname {d} \over \operatorname {d} x}\csc u=-\csc u\cot u}
Proof for the derivative of
{\displaystyle \sin u}
{\displaystyle y=\sin u}
{\displaystyle y\prime =\sin(u+\Delta u)}
{\displaystyle \Delta y=\sin(u+\Delta u)-\sin u}
In Trigonometry,
{\displaystyle \sin A-\sin B=2\sin {1 \over 2}(A-B)\cos {1 \over 2}(A+B)}
{\displaystyle A=u+\Delta u}
{\displaystyle B=u}
{\displaystyle \Delta y=2\cos \left(u+{\Delta u \over 2}\right)\ sin{\Delta u \over 2}}
{\displaystyle {\Delta y \over \Delta x}=\cos \left(u+{\Delta u \over 2}\right){\sin {\Delta u \over 2} \over {\Delta u \over 2}}}
{\displaystyle \Delta x}
approaches zero,
{\displaystyle \Delta u}
likewise approaches zero, and as
{\displaystyle \Delta u}
is in circular measure, the limit of
{\displaystyle {\sin {\Delta u \over 2} \over {\Delta u \over 2}}}
{\displaystyle {\operatorname {d} y \over \operatorname {d} x}=\cos u}
{\displaystyle {\operatorname {d} \over \operatorname {d} x}\cos u}
{\displaystyle {\operatorname {d} \over \operatorname {d} x}\cos u=\sin u\left(-{\operatorname {d} u \over \operatorname {d} x}\right)=-\sin u{\operatorname {d} u \over \operatorname {d} x}}
{\displaystyle {\operatorname {d} \over \operatorname {d} x}\tan u}
{\displaystyle \tan u={\sin u \over \cos u}}
{\displaystyle {\operatorname {d} \over \operatorname {d} x}\tan u={\cos u{\operatorname {d} \over \operatorname {d} x}\sin u-\sin u{\operatorname {d} \over \operatorname {d} x}\cos u \over \cos ^{2}u}}
{\displaystyle ={\cos ^{2}u{\operatorname {d} u \over \operatorname {d} x}+\sin ^{2}u{\operatorname {d} u \over \operatorname {d} x} \over \cos ^{2}u}={\operatorname {d} u \over \cos ^{2}u}}
{\displaystyle =\sec ^{2}u{\operatorname {d} u \over \operatorname {d} x}}
{\displaystyle {\operatorname {d} \over \operatorname {d} x}\sec u}
{\displaystyle {\operatorname {d} \over \operatorname {d} x}\tan u={\cos u{\operatorname {d} \over {d}x}\sin u-\sin u{\operatorname {d} \over \operatorname {d} x}\cos u \over \cos ^{2}u}}
{\displaystyle ={\cos ^{2}{\operatorname {d} u \over \operatorname {d} x}+\sin ^{2}u{\operatorname {d} u \over \operatorname {d} x} \over \cos ^{2}u}={{\operatorname {d} u \over \operatorname {d} x} \over \cos ^{2}u}}
{\displaystyle =\sec ^{2}u{\operatorname {d} u \over \operatorname {d} x}}
Retrieved from "https://en.wikibooks.org/w/index.php?title=High_School_Calculus/Derivatives_of_Trigonometric_Functions&oldid=2074159"
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Curve Fitting and Distribution Fitting - MATLAB & Simulink Example - MathWorks France
Choose Between Curve Fitting and Distribution Fitting
This example shows how to perform curve fitting and distribution fitting, and discusses when each method is appropriate.
Curve fitting and distribution fitting are different types of data analysis.
Use curve fitting when you want to model a response variable as a function of a predictor variable.
Use distribution fitting when you want to model the probability distribution of a single variable.
In the following experimental data, the predictor variable is time, the time after the ingestion of a drug. The response variable is conc, the concentration of the drug in the bloodstream. Assume that only the response data conc is affected by experimental error.
Suppose you want to model blood concentration as a function of time. Plot conc against time.
Assume that conc follows a two-parameter Weibull curve as a function of time. A Weibull curve has the form and parameters
y=c\left(x/a{\right)}^{\left(b-1\right)}{e}^{-\left(x/a{\right)}^{b}},
a
is a horizontal scaling,
b
is a shape parameter, and
c
is a vertical scaling.
Fit the Weibull model using nonlinear least squares.
Plot the Weibull curve onto the data.
The fitted Weibull model is problematic. fitnlm assumes the experimental errors are additive and come from a symmetric distribution with constant variance. However, the scatter plot shows that the error variance is proportional to the height of the curve. Furthermore, the additive, symmetric errors imply that a negative blood concentration measurement is possible.
A more realistic assumption is that multiplicative errors are symmetric on the log scale. Under that assumption, fit a Weibull curve to the data by taking the log of both sides. Use nonlinear least squares to fit the curve:
\mathrm{log}\left(y\right)=\mathrm{log}\left(c\right)+\left(b-1\right)\mathrm{log}\left(x/a\right)-\left(x/a{\right)}^{b}.
Add the new curve to the existing plot.
The model object nlModel2 contains estimates of precision. A best practice is to check the model's goodness of fit. For example, make residual plots on the log scale to check the assumption of constant variance for the multiplicative errors.
In this example, using the multiplicative errors model has little effect on the model predictions. For an example where the type of model has more of an impact, see Pitfalls in Fitting Nonlinear Models by Transforming to Linearity.
Statistics and Machine Learning Toolbox™ includes these functions for fitting models: fitnlm for nonlinear least-squares models, fitglm for generalized linear models, fitrgp for Gaussian process regression models, and fitrsvm for support vector machine regression models.
Curve Fitting Toolbox™ provides command line and graphical tools that simplify tasks in curve fitting. For example, the toolbox provides automatic choice of starting coefficient values for various models, as well as robust and nonparametric fitting methods.
Optimization Toolbox™ has functions for performing complicated types of curve fitting analyses, such as analyzing models with constraints on the coefficients.
The MATLAB® function polyfit fits polynomial models, and the MATLAB function fminsearch is useful in other kinds of curve fitting.
Suppose you want to model the distribution of electrical component lifetimes. The variable life measures the time to failure for 50 identical electrical components.
Visualize the data with a histogram.
Because lifetime data often follows a Weibull distribution, one approach might be to use the Weibull curve from the previous curve fitting example to fit the histogram. To try this approach, convert the histogram to a set of points (x,y), where x is a bin center and y is a bin height, and then fit a curve to those points.
Fitting a curve to a histogram, however, is problematic and usually not recommended.
The process violates basic assumptions of least-squares fitting. The bin counts are nonnegative, implying that measurement errors cannot be symmetric. Also, the bin counts have different variability in the tails than in the center of the distribution. Finally, the bin counts have a fixed sum, implying that they are not independent measurements.
If you fit a Weibull curve to the bar heights, you have to constrain the curve because the histogram is a scaled version of an empirical probability density function (pdf).
For continuous data, fitting a curve to a histogram rather than data discards information.
The bar heights in the histogram are dependent on the choice of bin edges and bin widths.
For many parametric distributions, maximum likelihood is a better way to estimate parameters because it avoids these problems. The Weibull pdf has almost the same form as the Weibull curve:
y=\left(b/a\right)\left(x/a{\right)}^{\left(b-1\right)}{e}^{-\left(x/a{\right)}^{b}}.
b/a
replaces the scale parameter
c
because the function must integrate to 1. To fit a Weibull distribution to the data using maximum likelihood, use fitdist and specify 'Weibull' as the distribution name. Unlike least squares, maximum likelihood finds a Weibull pdf that best matches the scaled histogram without minimizing the sum of the squared differences between the pdf and the bar heights.
Plot a scaled histogram of the data and superimpose the fitted pdf.
A best practice is to check the model's goodness of fit.
Although fitting a curve to a histogram is usually not recommended, the process is appropriate in some cases. For an example, see Fit Custom Distributions.
Statistics and Machine Learning Toolbox™ includes the function fitdist for fitting probability distribution objects to data. It also includes dedicated fitting functions (such as wblfit) for fitting parametric distributions using maximum likelihood, the function mle for fitting custom distributions without dedicated fitting functions, and the function ksdensity for fitting nonparametric distribution models to data.
Statistics and Machine Learning Toolbox additionally provides the Distribution Fitter app, which simplifies many tasks in distribution fitting, such as generating visualizations and diagnostic plots.
Functions in Optimization Toolbox™ enable you to fit complicated distributions, including those with constraints on the parameters.
The MATLAB® function fminsearch provides maximum likelihood distribution fitting.
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Health - Mango Markets
In v3, the health of an account is used to determine if an account can open a new position or be liquidated. There are two types of health:
initial health, used for opening new positions maintenance health, used for liquidations They are both calculated as a weighted sum of the assets minus the liabilities, but Maintenance health uses slightly larger weights for assets and slightly smaller weights for the liabilities. Zero is the threshold for both healths. If your Initial health is below falls below zero, you cannot open new positions and if your Maintenance health falls below zero you will be liquidated.
Health Calculation:
health = \sum\limits_{i} a_i \cdot p_i \cdot w^a_i - l_i \cdot p_i \cdot w^l_i \newline where \newline a_i-quantity\space asset \space i \newline p_i - price \space of \space i \newline w^a_i - asset \space weight \newline w^l_i - liability \space weight \newline l_i - quantity \space liability \space i
The Health Ratio which is shown in the trading page, is health rearranged: The sum of all Assets weighted divided by sum of all Liabs weighted - 1 The asset_weight applies a haircut to the value of the collateral in health calculations. The lower the asset weight, the less the asset counts towards collateral. Initial Leverage and Maintenance Leverage can be converted to the corresponding asset_weights with these calculations:
These values by asset are:\
Maint. Asset Weight
Maint. Liab. Weight
Init. Asset Weight
Init. Liab. Weight
For instance, the Maint. Asset Weight is used when calculating health to determine if the position is eligible for liquidation (edited)
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Arg max - Wikipedia
Find sources: "Arg max" – news · newspapers · books · scholar · JSTOR (October 2014) (Learn how and when to remove this template message)
As an example, both unnormalised and normalised sinc functions above have
{\displaystyle \operatorname {argmax} }
of {0} because both attain their global maximum value of 1 at x = 0.
The unnormalised sinc function (red) has arg min of {−4.49, 4.49}, approximately, because it has 2 global minimum values of approximately −0.217 at x = ±4.49. However, the normalised sinc function (blue) has arg min of {−1.43, 1.43}, approximately, because their global minima occur at x = ±1.43, even though the minimum value is the same.[1]
In mathematics, the arguments of the maxima (abbreviated arg max or argmax) are the points, or elements, of the domain of some function at which the function values are maximized.[note 1] In contrast to global maxima, which refers to the largest outputs of a function, arg max refers to the inputs, or arguments, at which the function outputs are as large as possible.
1.1 Arg min
Given an arbitrary set
{\displaystyle X}
, a totally ordered set
{\displaystyle Y}
, and a function,
{\displaystyle f\colon X\to Y}
{\displaystyle \operatorname {argmax} }
over some subset
{\displaystyle S}
{\displaystyle X}
{\displaystyle \operatorname {argmax} _{S}f:={\underset {x\in S}{\operatorname {arg\,max} }}\,f(x):=\{x\in S~:~f(s)\leq f(x){\text{ for all }}s\in S\}.}
{\displaystyle S=X}
{\displaystyle S}
is clear from the context, then
{\displaystyle S}
is often left out, as in
{\displaystyle {\underset {x}{\operatorname {arg\,max} }}\,f(x):=\{x~:~f(s)\leq f(x){\text{ for all }}s\in X\}.}
{\displaystyle \operatorname {argmax} }
{\displaystyle x}
{\displaystyle f(x)}
attains the function's largest value (if it exists).
{\displaystyle \operatorname {Argmax} }
may be the empty set, a singleton, or contain multiple elements.
In the fields of convex analysis and variational analysis, a slightly different definition is used in the special case where
{\displaystyle Y=[-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}}
are the extended real numbers.[2] In this case, i{\displaystyle f}
is identically equal to
{\displaystyle \infty }
{\displaystyle S}
{\displaystyle \operatorname {argmax} _{S}f:=\varnothing }
{\displaystyle \operatorname {argmax} _{S}\infty :=\varnothing }
) and otherwise
{\displaystyle \operatorname {argmax} _{S}f}
is defined as above, where in this case
{\displaystyle \operatorname {argmax} _{S}f}
can also be written as:
{\displaystyle \operatorname {argmax} _{S}f:=\left\{x\in S~:~f(x)=\inf {}_{S}f\right\}}
where it is emphasized that this equality involving
{\displaystyle \inf {}_{S}f}
holds only when
{\displaystyle f}
{\displaystyle \infty }
{\displaystyle S}
Arg min[edit]
The notion of
{\displaystyle \operatorname {argmin} }
{\displaystyle \operatorname {arg\,min} }
), which stands for argument of the minimum, is defined analogously. For instance,
{\displaystyle {\underset {x\in S}{\operatorname {arg\,min} }}\,f(x):=\{x\in S~:~f(s)\geq f(x){\text{ for all }}s\in S\}}
are points
{\displaystyle x}
{\displaystyle f(x)}
attains its smallest value. It is the complementary operator of
{\displaystyle \operatorname {arg\,max} }
{\displaystyle Y=[-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}}
are the extended real numbers, i{\displaystyle f}
{\displaystyle -\infty }
{\displaystyle S}
{\displaystyle \operatorname {argmin} _{S}f:=\varnothing }
{\displaystyle \operatorname {argmin} _{S}-\infty :=\varnothing }
{\displaystyle \operatorname {argmin} _{S}f}
is defined as above and moreover, in this case (o{\displaystyle f}
not identically equal to
{\displaystyle -\infty }
) it also satisfies:
{\displaystyle \operatorname {argmin} _{S}f:=\left\{x\in S~:~f(x)=\sup {}_{S}f\right\}.}
{\displaystyle f(x)}
{\displaystyle 1-|x|,}
{\displaystyle f}
attains its maximum value of
{\displaystyle 1}
only at the point
{\displaystyle x=0.}
{\displaystyle {\underset {x}{\operatorname {arg\,max} }}\,(1-|x|)=\{0\}.}
{\displaystyle \operatorname {argmax} }
operator is different from the
{\displaystyle \max }
operator. The
{\displaystyle \max }
operator, when given the same function, returns the maximum value of the function instead of the point or points that cause that function to reach that value; in other words
{\displaystyle \max _{x}f(x)}
is the element in
{\displaystyle \{f(x)~:~f(s)\leq f(x){\text{ for all }}s\in S\}.}
{\displaystyle \operatorname {argmax} ,}
max may be the empty set (in which case the maximum is undefined) or a singleton, but unlike
{\displaystyle \operatorname {argmax} ,}
{\displaystyle \operatorname {max} }
may not contain multiple elements:[note 2] for example, if
{\displaystyle f(x)}
{\displaystyle 4x^{2}-x^{4},}
{\displaystyle {\underset {x}{\operatorname {arg\,max} }}\,\left(4x^{2}-x^{4}\right)=\left\{-{\sqrt {2}},{\sqrt {2}}\right\},}
{\displaystyle {\underset {x}{\operatorname {max} }}\,\left(4x^{2}-x^{4}\right)=\{4\}}
because the function attains the same value at every element of
{\displaystyle \operatorname {argmax} .}
{\displaystyle M}
{\displaystyle f,}
{\displaystyle \operatorname {argmax} }
is the level set of the maximum:
{\displaystyle {\underset {x}{\operatorname {arg\,max} }}\,f(x)=\{x~:~f(x)=M\}=:f^{-1}(M).}
We can rearrange to give the simple identity[note 3]
{\displaystyle f\left({\underset {x}{\operatorname {arg\,max} }}\,f(x)\right)=\max _{x}f(x).}
If the maximum is reached at a single point then this point is often referred to as the
{\displaystyle \operatorname {argmax} ,}
{\displaystyle \operatorname {argmax} }
is considered a point, not a set of points. So, for example,
{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {arg\,max} }}\,(x(10-x))=5}
(rather than the singleton set
{\displaystyle \{5\}}
), since the maximum value of
{\displaystyle x(10-x)}
{\displaystyle 25,}
which occurs for
{\displaystyle x=5.}
[note 4] However, in case the maximum is reached at many points,
{\displaystyle \operatorname {argmax} }
needs to be considered a set of points.
{\displaystyle {\underset {x\in [0,4\pi ]}{\operatorname {arg\,max} }}\,\cos(x)=\{0,2\pi ,4\pi \}}
because the maximum value of
{\displaystyle \cos x}
{\displaystyle 1,}
which occurs on this interval for
{\displaystyle x=0,2\pi }
{\displaystyle 4\pi .}
On the whole real line
{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {arg\,max} }}\,\cos(x)=\left\{2k\pi ~:~k\in \mathbb {Z} \right\},}
so an infinite set.
Functions need not in general attain a maximum value, and hence the
{\displaystyle \operatorname {argmax} }
is sometimes the empty set; for example,
{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {arg\,max} }}\,x^{3}=\varnothing ,}
{\displaystyle x^{3}}
is unbounded on the real line. As another example,
{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {arg\,max} }}\,\arctan(x)=\varnothing ,}
{\displaystyle \arctan }
{\displaystyle \pm \pi /2.}
However, by the extreme value theorem, a continuous real-valued function on a closed interval has a maximum, and thus a nonempty
{\displaystyle \operatorname {argmax} .}
^ For clarity, we refer to the input (x) as points and the output (y) as values; compare critical point and critical value.
^ Due to the anti-symmetry of
{\displaystyle \,\leq ,}
a function can have at most one maximal value.
^ This is an identity between sets, more particularly, between subsets of
{\displaystyle Y.}
{\displaystyle x(10-x)=25-(x-5)^{2}\leq 25}
{\displaystyle x-5=0.}
^ "The Unnormalized Sinc Function Archived 2017-02-15 at the Wayback Machine", University of Sydney
^ a b c Rockafellar & Wets 2009, pp. 1–37.
arg min and arg max at PlanetMath.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Arg_max&oldid=1072034578"
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Revision as of 20:23, 17 November 2014 by NikosA (talk | contribs) (→Deriving Physical Quantities: Added reference to source for band parameters)
{\displaystyle {\frac {W}{m^{2}*sr*nm}}}
{\displaystyle L_{\lambda {\text{Pixel, Band}}}={\frac {K_{\text{Band}}*q_{\text{Pixel, Band}}}{\Delta \lambda _{\text{Band}}}}}
{\displaystyle L_{\lambda {\text{Pixel,Band}}}}
{\displaystyle K_{\text{Band}}}
{\displaystyle q_{\text{Pixel,Band}}}
{\displaystyle \Delta _{\lambda _{\text{Band}}}}
{\displaystyle \rho _{p}={\frac {\pi *L\lambda *d^{2}}{ESUN\lambda *cos(\Theta _{S})}}}
{\displaystyle \rho }
{\displaystyle \pi }
{\displaystyle L\lambda }
{\displaystyle d}
{\displaystyle Esun}
{\displaystyle cos(\theta _{s})}
{\displaystyle {\frac {W}{m^{2}*\mu m}}}
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Tessellate S.T.E.M.S. 2019 | Brilliant Math & Science Wiki
Tessellate S.T.E.M.S. 2019
Aditya Raut, Agnishom Chattopadhyay, Tessellate S.T.E.M.S. Mathematics, and
Tessellate S.T.E.M.S Physics
Tessellate STEMS Computer Science
Inspired by the success of the maiden edition last year, Chennai Mathematical Institute is back with the unique examination for high school and college students, S.T.E.M.S.
S.T.E.M.S. (Scholastic Test of Excellence in Mathematical Sciences), as a part of the college fest Tessellate, is a nationwide contest in Mathematics, Physics and Computer Science, which gives students an opportunity to show off their problem-solving skills, win exciting prizes and attend a 3-day camp at CMI.
The exam is one of its kind, you can attempt it from anywhere! Moreover, you can make use of books and online resources!
The camp features renowned mathematicians, physicists, and computer scientists from some of the best research institutes in India (such as CMI, ISI Kolkata, ISI Bangalore, IMSc, IISc, the IIT's). Students from all age groups have fair representation. It provides a great opportunity for the participants to interact with people at CMI and get an insight into academic life. The camp also features talks by students of the aforementioned institutes. Successful participants receive exciting prizes, along with certificates signed by some of the best academics in the country. The organizers will be providing the travel fare, food, and accommodations to selected candidates for the entire duration of the camp. Top 100 participants will be rewarded with a certificate of participation.
Register now and get ready for some science!
Sample Papers and Practice Problem Sets
Video Lectures for Preparation
All times are in Indian Standard Time (IST, i.e. GMT + 5:30)
12th January 2019 (12:00 pm to 03:00 pm) - S.T.E.M.S. Physics
12th January 2019 (04:00 pm to 07:00 pm) - S.T.E.M.S. Computer Science
13th January 2019 (12:00 pm to 06:00 pm) - S.T.E.M.S. Mathematics
Registrations close - 10th January 2019 (11:59 pm)
S.T.E.M.S. Camp at CMI - 8th February to 10th February 2019
Eligibility - Students in Class 8th-12th, undergraduate and graduate studies can participate.
The papers will be made available as soon as the exam commences and submissions will be accepted via email.
The format of the exam paper follows after the name of every subject. Based on their academic year, students are divided into the following sections:
S.T.E.M.S. Mathematics - (15 Objective + 6 Subjective)
Section A - Class 10 and below
Section B - Class 11, Class 12, Undergraduate 1st year
Section C - Undergraduate 2nd year and above
S.T.E.M.S. Physics - (10 Objective + 3 Subjective)
S.T.E.M.S. Computer Science - (10 Objective + 3 Subjective)
Section B - Undergraduate 1st year and above
50
candidates with exceptional performance among all the examinees will be chosen for the camp at Chennai Mathematical Institute in February.
The solutions have to be submitted in electronic formats, scanned copies or clear photographs of the answer sheets. They have to be mailed to tessellate.cmi@gmail.com from your registered email ID strictly before the ending time.
Students are allowed to refer to books and online resources to solve the problems.
The problems should not be uploaded on any forums or websites for discussion during exam time. Any case of misconduct will not be shown the slightest compassion.
Your solutions must be strictly original. There might be an interview for confirmation after the selection is made. In case of any discrepancies, the submission of the student in question will be invalidated.
Submissions made after the deadline will not be accepted.
The examination fee is ₹100 per subject.
Registration is hassle-free. Visit our website tessellate.cmi.ac.in/stems for details.
Enter all the essential details during the online registration page.
For any clarifications required, you can reach us at tessellate.cmi@gmail.com
Alternatively, in case of any queries you may contact:
Aditya Raut: (+91) 9922793530 (adityaraut@cmi.ac.in)
Ashwani Anand: (+91) 9905913014 (ashwani@cmi.ac.in)
Soham Chakraborty: (+91) 9884232190 (sochak@cmi.ac.in)
Srijan Ghosh: (+91) 9433777622 (srijang@cmi.ac.in)
Sarvesh Bandhaokar: (+91) 9405956066 (bandhaokar@cmi.ac.in)
Stay tuned for the updates, the first practice sets for each category will be uploaded on 14th October 2018.
Tessellate S.T.E.M.S (2019) - Mathematics - Category A - Sample Paper
Tessellate S.T.E.M.S (2019) - Mathematics - Category B - Sample Paper
Tessellate S.T.E.M.S (2019) - Mathematics - Category C - Sample Paper
Tessellate S.T.E.M.S (2019) - Mathematics - Category A - Set 1
Tessellate S.T.E.M.S (2019) - Mathematics - Category B - Set 1
Tessellate S.T.E.M.S (2019) - Mathematics - Category C - Set 1
Tessellate S.T.E.M.S (2019) - Computer Science - School - Sample Paper
Tessellate S.T.E.M.S (2019) - Computer Science - College - Sample Paper
Tessellate S.T.E.M.S (2019) - Computer Science - School - Set 1
Tessellate S.T.E.M.S (2019) - Computer Science - College - Set 1
Tessellate S.T.E.M.S (2019) - Physics - Category A - Sample Paper
Tessellate S.T.E.M.S (2019) - Physics - Category B - Sample Paper
Tessellate S.T.E.M.S (2019) - Physics - Category C - Sample Paper
Tessellate S.T.E.M.S (2019) - Physics - Category A - Set 1
Tessellate S.T.E.M.S (2019) - Physics - Category B - Set 1
Tessellate S.T.E.M.S (2019) - Physics - Category C - Set 1
As a part of our campaign, lectures are organized at Chennai Mathematical Institute's campus in Siruseri. Visit our webpage tessellate.cmi.ac.in/stems for details.
The lecture series at Chennai Mathematical Institue begins on the 14th October 2018.
Visit our YouTube channel Tessellate CMI to access the videos of these lectures.
Basic Counting (Rule of Sum, Rule of Product, Combinations, Permutations, Principle of Inclusion-Exclusion)
Induction and Proof by Contradiction
Elementary Recurrence Relations and Characteristic Equations
Generating Functions and Binomial Theorem
Linear Equations, Quadratic Equations
Polynomials over known rings (
\mathbb{Z,Q,R}
\mathbb{C}
Classical Inequalities (AM-GM, Cauchy-Schwartz, Rearrangement, Schur's Inequality)
Exponents, Logarithms and Trigonometric Functions
Complex Numbers (De-Moivre, Polar Coordinates, Conjugates, and basic properties)
Sequence and Series (Arithmetic Progressions, Geometric Progression, Harmonic Progression etc.)
Euclidean Geometry (Triangle Geometry, Cyclic Quadrilaterals, Radical Axis, Geometric Transformations)
Coordinate Geometry (Distance Formula, Equations of Straight Lines, Equation of Circles)
Conic Sections (Equations, Geometric Properties)
Trigonometry Trigonometry (Basic properties of trigonometric functions, identities)
Modular Congruences (Euler's Theorem, Fermat's Little Theorem, Wilson's Theorem, Chinese Remainder Remainder Theorem may be helpful.)
Arithmetic Functions (Totient, Divisor, Sum of Divisors, Mobius Function)
Basics of Set Theory (Set union, intersection, symmetric difference)
Basics of Probability (Conditional Probability, Bayes' Theorem, Binomial Trials, Expected Value)
In addition to the syllabus of section A, the following topics -
Integrals, Applications of Integrals
Coordinate Geometry (Equations of Conic Sections)
Basics of Linearity of Expectation
Advanced knowledge of all concepts mentioned in the high school syllabus
Elementary knowledge of Forms (Bilinear Forms, Skew Symmetric Forms, etc.)
\big(
Functions from
\mathbb{R}^n \to \mathbb{R}^m
, their derivatives, and inverse function theorem (not mandatory) might be useful.
\big)
(
Matrix Groups, Cauchy and Sylow Theorems, Cayley's Theorems, Permutations, Finite Abelian Groups (not mandatory), Isomorphism Theorems
)
Ring Theory (Basics)
Field Theory (Basics)
Advanced Combinatorial Concepts
Probability Distribution Function (Bernoulli Distribution, Binomial Distribution, Poisson Distribution, Normal Distribution, Uniform Distribution, etc.)
Uniform and Non-uniform Motion along a Straight Line
Pressure in Fluids, Pascal's Law
Conduction, Elementary Concepts of Convection and Radiation
Resistance of System of Resistors (Series and Parallel)
Magnetic Fields and Field Lines
Magnetic Field - Right-hand Thumb Rule
Friction (Static and Dynamic)
Linear and Angular Harmonic Motions
Bernoulli's Theorems and its Applications
Gauss's Law and its Application in Simple Cases
Electric Current, Ohm's Law, Series and Parallel Arrangements of Resistors and Cells, Kirchoff's Laws (and Simple Applications)
Electromagnetic Induction: Faraday's Law, Lenz's Law, RC, LC, and RL Circuits
Conduction in 1 Dimension, Elementary concepts of Convection and Radiation
Black Body Radiation (Absorptive and Emissive Powers): Kirchoff's Law, Wein's Displacement Law, Stefan Law
Wave Nature of Light: Huygens Principle, Interference
Law of Radioactive Decay, Decay Constant, Half-life and Mean Life, Binding Energy and its Calculation, Fission and Fusion Processes
Bohr's Theory of Hydrogen-like Atoms
Newtonian Mechanics, Lagrangian Mechanics, Hamiltonian Mechanics
Special Relativity (Time Dilation, Length Contraction, Lorentz Transformation)
Gauss's Law, Coulomb's Law, Application of Gauss's Law in the Presence of Symmetries
Currents and AC and DC Circuits
Solution of Laplace's Equations in Cartesian, Spherical, and Cylindrical Coordinates
Electromagnetic Waves and Poynting's Theorem
Heisenberg's Formulation, Schrodinger's Formulation
\frac{1}{2}
Angular Momentum Quantization and Addition
Perturbation Theory (Basics)
Superposition, Diffraction
Thermodynamic Processes, Equations of State
Ideal Gases, Kinetic Theory
The objective of the exam is to test the student on their computational, algorithmic, logical thinking abilities and theoretical aspects of computation. Specific details about hardware architecture, operating systems, software systems, web technologies, programming languages, etc. will not be asked.
Everything included in IOI syllabus
The main focus will be on the following aspects:
Systematically following, simulating and reasoning about sets of instructions, protocols, structures, etc.
Understanding the correctness of algorithms
Assessing performance of algorithms
Reasoning about discrete structures
Reasoning about combinatorial games
Understanding implications of logical statements
Graph algorithms (connectivity, spanning trees, matchings, flows etc.)
Number-theoretic algorithms (primality testing, factorization etc.)
Divide and conquer, dynamic programming, greedy algorithms, and other common techniques
Basic running time analysis
Basic complexity classes (P, NP, P-space etc.)
Interactive proofs, probabilistically checkable proofs
DFA/NFA and regular languages
Context-free grammars and pushdown automata
Turing machines / Oracle Turing machines
Basic programming in a language of choice
Comprehensive understanding of algorithms and algorithmic paradigms such as greedy algorithms, dynamic programming, divide & conquer, and introductory graph algorithms. A preliminary knowledge of analysis of these algorithms is essential.
Understanding of data structures and various discrete structures such as graphs, trees, heaps, stacks, and queues.
An understanding of finite state machines, pushdown automata, and Turing machines, along with their properties and representations including grammars and computation models.
An understanding of computation in terms of complexity and decidability.
To know more about the previous editions of S.T.E.M.S. visit the Brilliant wiki.
You can also access the practice problem sets and actual exam papers of S.T.E.M.S. 2018 in the wiki.
Cite as: Tessellate S.T.E.M.S. 2019. Brilliant.org. Retrieved from https://brilliant.org/wiki/tessellate-stems-2019/
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Create Simulink Environment and Train Agent - MATLAB & Simulink - MathWorks Deutschland
This example shows how to convert the PI controller in the watertank Simulink® model to a reinforcement learning deep deterministic policy gradient (DDPG) agent. For an example that trains a DDPG agent in MATLAB®, see Train DDPG Agent to Control Double Integrator System.
The original model for this example is the water tank model. The goal is to control the level of the water in the tank. For more information about the water tank model, see watertank Simulink Model (Simulink Control Design).
Modify the original model by making the following changes:
Insert the RL Agent block.
Connect the observation vector
{\left[\begin{array}{ccc}\int \mathit{e}\text{\hspace{0.17em}}\mathrm{dt}& \mathit{e}& \mathit{h}\end{array}\right]}^{\mathit{T}\text{\hspace{0.17em}}}
\mathit{h}
is the height of the tank,
\mathit{e}=\mathit{r}-\mathit{h}
\mathit{r}
Set up the reward
\mathrm{reward}=10\left(|\mathit{e}|<0.1\right)-1\left(|\mathit{e}|\ge 0.1\right)-100\left(\mathit{h}\le 0||\mathit{h}\ge 20\right)
Configure the termination signal such that the simulation stops if
\mathit{h}\le 0
\mathit{h}\ge 20
The resulting model is rlwatertank.slx. For more information on this model and the changes, see Create Simulink Reinforcement Learning Environments.
Action and observation signals that the agent uses to interact with the environment. For more information, see rlNumericSpec and rlFiniteSetSpec.
Define the observation specification obsInfo and action specification actInfo.
numActions = actInfo.Dimension(1);
Build the environment interface object.
env = rlSimulinkEnv('rlwatertank','rlwatertank/RL Agent',...
Set a custom reset function that randomizes the reference values for the model.
Given observations and actions, a DDPG agent approximates the long-term reward using a critic value function representation. To create the critic, first create a deep neural network with two inputs, the observation and action, and one output. For more information on creating a deep neural network value function representation, see Create Policies and Value Functions.
fullyConnectedLayer(25,'Name','CriticStateFC2')];
fullyConnectedLayer(25,'Name','CriticActionFC1')];
criticOpts = rlRepresentationOptions('LearnRate',1e-03,'GradientThreshold',1);
Create the critic representation using the specified deep neural network and options. You must also specify the action and observation specifications for the critic, which you obtain from the environment interface. For more information, see rlQValueRepresentation.
critic = rlQValueRepresentation(criticNetwork,obsInfo,actInfo,'Observation',{'State'},'Action',{'Action'},criticOpts);
Given observations, a DDPG agent decides which action to take using an actor representation. To create the actor, first create a deep neural network with one input, the observation, and one output, the action.
Construct the actor in a similar manner to the critic. For more information, see rlDeterministicActorRepresentation.
fullyConnectedLayer(3, 'Name','actorFC')
actor = rlDeterministicActorRepresentation(actorNetwork,obsInfo,actInfo,'Observation',{'State'},'Action',{'Action'},actorOptions);
'DiscountFactor',1.0, ...
Run each training for at most 5000 episodes. Specify that each episode lasts for at most ceil(Tf/Ts) (that is 200) time steps.
Stop training when the agent receives an average cumulative reward greater than 800 over 20 consecutive episodes. At this point, the agent can control the level of water in the tank.
'ScoreAveragingWindowLength',20, ...
load('WaterTankDDPG.mat','agent')
simOpts = rlSimulationOptions('MaxSteps',maxsteps,'StopOnError','on');
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Time Series Regression I: Linear Models - MATLAB & Simulink Example - MathWorks Australia
Multiple Linear Models
This example introduces basic assumptions behind multiple linear regression models. It is the first in a series of examples on time series regression, providing the basis for all subsequent examples.
Time series processes are often described by multiple linear regression (MLR) models of the form:
{y}_{t}={X}_{t}\beta +{e}_{t},
{y}_{t}
is an observed response and
{X}_{t}
includes columns for contemporaneous values of observable predictors. The partial regression coefficients in
\beta
represent the marginal contributions of individual predictors to the variation in
{y}_{t}
when all of the other predictors are held fixed.
{e}_{t}
is a catch-all for differences between predicted and observed values of
{y}_{t}
. These differences are due to process fluctuations (changes in
\beta
), measurement errors (changes in
{X}_{t}
), and model misspecifications (for example, omitted predictors or nonlinear relationships between
{X}_{t}
{y}_{t}
). They also arise from inherent stochasticity in the underlying data-generating process (DGP), which the model attempts to represent. It is usually assumed that
{e}_{t}
is generated by an unobservable innovations process with stationary covariance
{\Omega }_{T}=Cov\left(\left\{{e}_{1},...,{e}_{T}\right\}\right),
for any time interval of length
T
. Under some further basic assumptions about
{X}_{t}
{e}_{t}
, and their relationship, reliable estimates of
\beta
are obtained by ordinary least squares (OLS).
As in other social sciences, economic data are usually collected by passive observation, without the aid of controlled experiments. Theoretically relevant predictors may need to be replaced by practically available proxies. Economic observations, in turn, may have limited frequency, low variability, and strong interdependencies.
These data shortcomings lead to a number of issues with the reliability of OLS estimates and the standard statistical techniques applied to model specification. Coefficient estimates may be sensitive to data measurement errors, making significance tests unreliable. Simultaneous changes in multiple predictors may produce interactions that are difficult to separate into individual effects. Observed changes in the response may be correlated with, but not caused by, observed changes in the predictors.
Assessing model assumptions in the context of available data is the goal of specification analysis. When the reliability of a model becomes suspect, practical solutions may be limited, but a thorough analysis can help to identify the source and degree of any problems.
This is the first in a series of examples that discuss basic techniques for specifying and diagnosing MLR models. The series also offers some general strategies for addressing the specific issues that arise when working with economic time series data.
Classical linear model (CLM) assumptions allow OLS to produce estimates
\underset{}{\overset{ˆ}{\beta }}
with desirable properties [3]. The fundamental assumption is that the MLR model, and the predictors selected, correctly specify a linear relationship in the underlying DGP. Other CLM assumptions include:
{X}_{t}
is full rank (no collinearity among the predictors).
{e}_{t}
{X}_{s}
s
(strict exogeneity of the predictors).
{e}_{t}
is not autocorrelated (
{\Omega }_{T}
is diagonal).
{e}_{t}
is homoscedastic (diagonal entries in
{\Omega }_{T}
{\sigma }^{2}
ϵ=\underset{}{\overset{ˆ}{\beta }}-\beta
is the estimation error. The bias of the estimator is
E\left[ϵ\right]
and the mean-squared error (MSE) is
E\left[{ϵ}^{\prime }ϵ\right]
. The MSE is the sum of the estimator variance and the square of the bias, so it neatly summarizes two important sources of estimator inaccuracy. It should not be confused with regression MSE, concerning model residuals, which is sample dependent.
All estimators are limited in their ability minimize the MSE, which can never be smaller than the Cramér-Rao lower bound [1]. This bound is achieved asymptotically (that is, as the sample size grows larger) by the maximum likelihood estimator (MLE). However, in finite samples, and especially in the relatively small samples encountered in economics, other estimators may compete with the MLE in terms of relative efficiency, that is, in terms of the achieved MSE.
Under the CLM assumptions, the Gauss-Markov theorem says that the OLS estimator
\underset{}{\overset{ˆ}{\beta }}
is BLUE:
B est (minimum variance)
L inear (linear function of the data)
U nbiased (
E\left[\underset{}{\overset{ˆ}{\beta }}\right]=\beta
E stimator of the coefficients in
\beta
BEST adds up to a minimum MSE among linear estimators. Linearity is important because the theory of linear vector spaces can be applied to the analysis of the estimator (see, for example [5]).
If the innovations
{e}_{t}
are normally distributed,
\underset{}{\overset{ˆ}{\beta }}
will also be normally distributed. In that case, reliable
t
F
tests can be carried out on the coefficient estimates to assess predictor significance, and confidence intervals can be constructed to describe estimator variance using standard formulas. Normality also allows
\underset{}{\overset{ˆ}{\beta }}
to achieve the Cramér-Rao lower bound (it becomes efficient), with estimates identical to the MLE.
Regardless of the distribution of
{e}_{t}
, the Central Limit theorem assures that
\underset{}{\overset{ˆ}{\beta }}
will be approximately normally distributed in large samples, so that standard inference techniques related to model specification become valid asymptotically. However, as noted earlier, samples of economic data are often relatively small, and the Central Limit theorem cannot be relied upon to produce a normal distribution of estimates.
Static econometric models represent systems that respond exclusively to current events. Static MLR models assume that the predictors forming the columns of
{X}_{t}
are contemporaneous with the response
{y}_{t}
. Evaluation of CLM assumptions is relatively straightforward for these models.
By contrast, dynamic models use lagged predictors to incorporate feedback over time. There is nothing in the CLM assumptions that explicitly excludes predictors with lags or leads. Indeed, lagged exogenous predictors
{x}_{t-k}
, free from interactions with the innovations
{e}_{t}
, do not, in themselves, affect the Gauss-Markov optimality of OLS estimation. If predictors include proximate lags
{x}_{t-k}
{x}_{t-k-1}
{x}_{t-k-2}
, ..., however, as economic models often do, then predictor interdependencies are likely to be introduced, violating the CLM assumption of no collinearity, and producing associated problems for OLS estimation. This issue is discussed in the example Time Series Regression II: Collinearity and Estimator Variance.
When predictors are endogenous, determined by lagged values of the response
{y}_{t}
(autoregressive models), the CLM assumption of strict exogeneity is violated through recursive interactions between the predictors and the innovations. In this case other, often more serious, problems of OLS estimation arise. This issue is discussed in the example Time Series Regression VIII: Lagged Variables and Estimator Bias.
Violations of CLM assumptions on
{\Omega }_{T}
(nonspherical innovations) are discussed in the example Time Series Regression VI: Residual Diagnostics.
Violations of CLM assumptions do not necessarily invalidate the results of OLS estimation. It is important to remember, however, that the effect of individual violations will be more or less consequential, depending on whether or not they are combined with other violations. Specification analysis attempts to identify the full range of violations, assess the effects on model estimation, and suggest possible remedies in the context of modeling goals.
Consider a simple MLR model of credit default rates. The file Data_CreditDefaults.mat contains historical data on investment-grade corporate bond defaults, as well as data on four potential predictors for the years 1984 to 2004:
X0 = Data(:,1:4); % Initial predictor set (matrix)
X0Tbl = DataTable(:,1:4); % Initial predictor set (tabular array)
predNames0 = series(1:4); % Initial predictor set names
T0 = size(X0,1); % Sample size
y0 = Data(:,5); % Response data
respName0 = series{5}; % Response data name
The potential predictors, measured for year t, are:
AGE Percentage of investment-grade bond issuers first rated 3 years ago. These relatively new issuers have a high empirical probability of default after capital from the initial issue is expended, which is typically after about 3 years.
BBB Percentage of investment-grade bond issuers with a Standard & Poor's credit rating of BBB, the lowest investment grade. This percentage represents another risk factor.
CPF One-year-ahead forecast of the change in corporate profits, adjusted for inflation. The forecast is a measure of overall economic health, included as an indicator of larger business cycles.
SPR Spread between corporate bond yields and those of comparable government bonds. The spread is another measure of the risk of current issues.
The response, measured for year t+1, is:
IGD Default rate on investment-grade corporate bonds
As described in [2] and [4], the predictors are proxies, constructed from other series. The modeling goal is to produce a dynamic forecasting model, with a one-year lead in the response (equivalently, a one-year lag in the predictors).
We first examine the data, converting the dates to a datetime vector so that the utility function recessionplot can overlay bands showing relevant dips in the business cycle:
% Convert dates to datetime vector:
dt = datetime(string(dates),'Format','yyyy');
% Plot potential predictors:
plot(dt,X0,'LineWidth',2)
title('{\bf Potential Predictors}')
% Plot response:
plot(dt,y0,'k','LineWidth',2);
plot(dt,y0-detrend(y0),'m--')
legend(respName0,'Linear Trend','Location','NW')
We see that BBB is on a slightly different scale than the other predictors, and trending over time. Since the response data is for year t + 1, the peak in default rates actually follows the recession in t = 2001.
The predictor and response data can now be assembled into an MLR model, and the OLS estimate of
\underset{}{\overset{ˆ}{\beta }}
can be found with the MATLAB backslash (\) operator:
% Add intercept to model:
X0I = [ones(T0,1),X0]; % Matrix
X0ITbl = [table(ones(T0,1),'VariableNames',{'Const'}),X0Tbl]; % Table
Estimate = X0I\y0
Estimate = 5×1
Alternatively, the model can be examined with LinearModel object functions, which provide diagnostic information and many convenient options for analysis. The function fitlm is used to estimate the model coefficients in
\underset{}{\overset{ˆ}{\beta }}
from the data. It adds an intercept by default. Passing in the data in the form of a tabular array, with variable names, and the response values in the last column, returns a fitted model with standard diagnostic statistics:
M0 = fitlm(DataTable)
There remain many questions to be asked about the reliability of this model. Are the predictors a good subset of all potential predictors of the response? Are the coefficient estimates accurate? Is the relationship between predictors and response, indeed, linear? Are model forecasts dependable? In short, is the model well-specified and does OLS do a good job fitting it to the data?
Another LinearModel object function, anova, returns additional fit statistics in the form of a tabular array, useful for comparing nested models in a more extended specification analysis:
ANOVATable = anova(M0)
ANOVATable=5×5 table
SumSq DF MeanSq F pValue
________ __ _________ ______ _________
AGE 0.019457 1 0.019457 3.3382 0.086402
BBB 0.014863 1 0.014863 2.55 0.12985
CPF 0.089108 1 0.089108 15.288 0.0012473
SPR 0.010435 1 0.010435 1.7903 0.1996
Error 0.09326 16 0.0058287
Model specification is one of the fundamental tasks of econometric analysis. The basic tool is regression, in the broadest sense of parameter estimation, used to evaluate a range of candidate models. Any form of regression, however, relies on certain assumptions, and certain techniques, which are almost never fully justified in practice. As a result, informative, reliable regression results are rarely obtained by a single application of standard procedures with default settings. They require, instead, a considered cycle of specification, analysis, and respecification, informed by practical experience, relevant theory, and an awareness of the many circumstances where poorly considered statistical evidence can confound sensible conclusions.
Exploratory data analysis is a key component of such analyses. The basis of empirical econometrics is that good models arise only through interaction with good data. If data are limited, as is often the case in econometrics, analysis must acknowledge the resulting ambiguities, and help to identify a range of alternative models to consider. There is no standard procedure for assembling the most reliable model. Good models emerge from the data, and are adaptable to new information.
Subsequent examples in this series consider linear regression models, built from a small set of potential predictors and calibrated to a rather small set of data. Still, the techniques, and the MATLAB toolbox functions considered, are representative of typical specification analyses. More importantly, the workflow, from initial data analysis, through tentative model building and refinement, and finally to testing in the practical arena of forecast performance, is also quite typical. As in most empirical endeavors, the process is the point.
[1] Cramér, H. Mathematical Methods of Statistics. Princeton, NJ: Princeton University Press, 1946.
[2] Helwege, J., and P. Kleiman. "Understanding Aggregate Default Rates of High Yield Bonds." Federal Reserve Bank of New York Current Issues in Economics and Finance. Vol. 2, No. 6, 1996, pp. 1–6.
[3] Kennedy, P. A Guide to Econometrics. 6th ed. New York: John Wiley & Sons, 2008.
[5] Strang, G. Linear Algebra and Its Applications. 4th ed. Pacific Grove, CA: Brooks Cole, 2005.
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The radical expression \sqrt[n]{a} represents the ? root of a.
The radical expression \sqrt[n]{a} represents the ? root of a. The number
\sqrt[n]{a}
represents the ? root of a. The number n is called the ?.
In algebra we have numbers involving radicals and exponents. A radical expression is an expression containing square roots. An equation which contains radical expressions with variables is called radical equation.
An exponent is the expression of the form
{a}^{n}
. In general we use both form of expressions to solve exponent and radical equations.
The radical expression and the exponents are very important parts of algebra. To find the solution of an expression or an equation we frequently use the conversion formulas of radical to exponent and vice versa .
In the given Radical form
\sqrt[n]{a}
, it represents the n-th root of a variable a. The value or any variable term inside the square root symbol is called radicand. The value n is called the index of the radical .
For square root we have the index 2 , but for square root we will not mention index.
For cube root we have index 3 , and it is written as "
\sqrt{3}
Hence in general "
\sqrt{n}
" is of index n.
Hence answer of the first blank is to be filled with nth root.
And the answer of the second blank is to be filled with index.
{12.0}^{\circ }
Magnesium (Mg) reacts with chlorine gas
\left(C{l}_{2}\right)
to produce magnesium chloride
\left(MgC{l}_{2}\right)
. How many grams of Mg will react with 2 moles of
C{l}_{2}
Mg\left(s\right)+C{l}_{2}\left(g\right)=MgC{l}_{2}\left(s\right)
The government of a large city needs to determine whether the citys
A scalloped hammerhead shark swims at a steady speed of 1.0 m/sm/s with its 83 cm -cm-wide head perpendicular to the earth's
52\mu T
magnetic field.
What is the magnitude of the emf induced between the two sides of the shark's head?
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Direct Torque Control of Permanent Magnet Synchronous Motor Based on Sliding Mode Variable Structure
Abstract: In order to solve the problem of large torque and flux ripple in the traditional direct torque control of permanent magnet synchronous motor, Super-Twisting sliding mode variable structure control strategy is adopted. Under the dl coor-dinate system, the torque and flux controller is designed. In the direct torque control, the problem of speed overshoot and considerable fluctuation during PI control is adopted. The synovial speed controller is designed, and the space vector pulse width modulation technology (SVPWM) is adopted to stabilize the inverter switching frequency. The simulation results demonstrate that the method can effectively reduce the torque and flux linkage and accelerate the system response.
Keywords: Permanent Magnet Synchronous Motor, Direct Torque Control, Sliding Mode Variable Structure, Space Vector Pulse Width Modulation Technology
Permanent magnet synchronous motor has the characteristics of small size, lightweight, high power factor and high efficiency, and is widely used in the field of robots and electric vehicles [1] . Direct torque control of permanent magnet synchronous motor is widely used in high performance servo applications due to its high working efficiency, small parameter dependence on the motor, strong robustness, good dynamic performance and simple structure control [2] . In the traditional direct torque control, the hysteresis controller is used to control the torque and flux linkage. The torque and flux linkage are large, the switching frequency is unstable, and the PI control is mainly used in the speed control. When the system has speed changes and load disturbances, it is impossible to achieve fast and no overshoot tracking target speed.
A lot of research has been conducted by domestic and foreign scholars on the problems existing in traditional direct torque control. Literature [3] introduces zero voltage vector and vector subdivision to improve the voltage vector switch table, which reduces the flux linkage and torque ripple to a certain extent. The literature [4] uses space vector pulse width modulation technologies to reduce the inverter. On the basis of switching frequency, many scholars combine intelligent control with traditional direct torque control. The literature [5] utilizes fuzzy control to optimize the selection of voltage space vector, but the fuzzy rules are complex and rely on experience, sliding mode variable structure. Control is applicable to the direct torque control of permanent magnet synchronous motor due to its robustness, rapid response, and insensitivity to external disturbances [6] .
In this paper, the super-spiral sliding mode variable structure control is used to design the flux linkage and torque controller. The velocity controller is designed by using the sliding film control method based on the approach law. The system designed in this paper is compared with the traditional direct torque control system. It shows that the method adopted in this paper can effectively reduce the ripple of flux linkage and torque, and accelerate the speed response and anti-disturbance capability.
2. Mathematical Model of PMSM
In order to simplify the analysis, the following assumptions are often made in the mathematical modeling of permanent magnet synchronous motors:
1) Ignore the saturation of the iron core inside the motor;
2) Excluding the eddy current and hysteresis loss when the motor is running;
3) The stator winding current is a three-phase sinusoidal current, and the induced electromotive force of the stator armature winding is also a sine wave [7] .
2.1. Mathematical Model of Two-Phase Stationary Coordinate System αβ
The stator voltage equation is:
\left\{\begin{array}{l}{u}_{\alpha }={L}_{s}\frac{\text{d}{i}_{\alpha }}{\text{d}t}+{R}_{s}{i}_{\alpha }-{\omega }_{r}{\psi }_{f}\mathrm{sin}{\theta }_{r}\\ {u}_{\beta }={L}_{s}\frac{\text{d}{i}_{\beta }}{\text{d}t}+{R}_{s}{i}_{\beta }+{\omega }_{r}{\psi }_{f}\mathrm{cos}{\theta }_{r}\end{array}
{u}_{\alpha },{u}_{\beta },{i}_{\alpha },{i}_{\beta }
are components of the stator voltage vector and the current vector on the α and β axes;
{R}_{s},{L}_{s}
are stator resistance and stator inductance;
{\omega }_{r}
is the motor angular velocity;
{\psi }_{f}
is a permanent magnet flux linkage;
{\theta }_{r}
is the rotor position angle.
The stator flux linkage equation is [8] :
\left\{\begin{array}{l}{\psi }_{\alpha }=\int \left({u}_{\alpha }-{R}_{s}{i}_{\alpha }\right)\text{d}t\\ {\psi }_{\beta }=\int \left({u}_{\beta }-{R}_{s}{i}_{\beta }\right)\text{d}t\end{array}
The electromagnetic torque equation is:
{T}_{e}=\frac{3}{2}p\left({\psi }_{\alpha }{i}_{\beta }-{\psi }_{\beta }{i}_{\alpha }\right)
The motor motion equation is:
\frac{\text{d}{\omega }_{m}}{\text{d}t}=\frac{p}{J}\left({T}_{e}-{T}_{L}\right)
where: p is the pole number of the motor;
J is the moment of inertia;
TL is the load torque.
2.2. Mathematical Model under Rotating Coordinate System dq
\left\{\begin{array}{l}{u}_{d}=\frac{\text{d}{\psi }_{d}}{\text{d}t}+{R}_{s}{i}_{d}-{\omega }_{r}{\psi }_{q}\\ {u}_{q}=\frac{\text{d}{\psi }_{q}}{\text{d}t}+{R}_{s}{i}_{q}+{\omega }_{r}{\psi }_{d}\end{array}
The stator flux linkage equation is:
\left\{\begin{array}{l}{\psi }_{d}={L}_{d}{i}_{d}+{\psi }_{f}\\ {\psi }_{q}={L}_{q}{i}_{q}\end{array}
The electromagnetic torque equation is [9] :
{T}_{e}=\frac{3}{2}p{i}_{q}\left[{i}_{d}\left({L}_{d}-{L}_{q}\right)+{\psi }_{f}\right]
{u}_{d},{u}_{q}
are the components of the stator voltage on the dq axis;
{i}_{d},{i}_{q}
are the components of the stator current on the dq axis;
{\psi }_{d},{\psi }_{q}
are the components of the stator flux linkage on the dq axis;
{L}_{d},{L}_{q}
are the inductances on the dq axis;
{\psi }_{f}
is the stator flux linkage.
3. PMSM Direct Torque Control Based on Sliding Mode Control
3.1. Super-Slip Sliding Mode Variable Structure Control Strategy
Sliding mode variable structure control has discontinuity, the system structure follows the switching characteristics of the time change, the system parameters are independent of the outside world, the response speed is fast, and it has good robustness. The super-spiral algorithm-based synovial controller can effectively eliminate the pulsation. The formula of the super-helical control algorithm is as follows:
\left\{\begin{array}{l}u={K}_{P}{|s|}^{r}\mathrm{sgn}\left(s\right)+{u}_{1}\\ \frac{\text{d}{u}_{1}}{\text{d}t}={K}_{I}\mathrm{sgn}\left(s\right)\end{array}
{K}_{P}
{K}_{I}
are gains.
The sufficient conditions for the stability of the control system are:
\left\{\begin{array}{l}{K}_{P}\ge \frac{{A}_{M}}{{B}_{m}}\\ {K}_{I}\ge \frac{4{A}_{M}}{{B}_{m}^{2}}\cdot \frac{{B}_{M}\left({K}_{P}+{A}_{M}\right)}{{B}_{m}\left({K}_{P}-{A}_{M}\right)}\\ 0\le r\le 0.5\end{array}
{B}_{m}\le B\le {B}_{M}
{A}_{M}\ge |A|
And A, B meets:
\frac{{\text{d}}^{2}y}{\text{d}{t}^{2}}=A\left(x,t\right)+B\left(x,t\right)\frac{\text{d}u}{\text{d}x}
Sliding mode variable structure control has discontinuity, the system structure follows the switching characteristics of the time change, the system parameters are independent of the outside world, the response speed is fast, and it has good robustness. The super-spiral algorithm-based synovial controller can effectively eliminate the pulsation. The formula of the super-helical control algorithm is as follows [10] :
Magnetic chain controller:
\left\{\begin{array}{l}{u}_{d}={K}_{P1}{|{S}_{\psi }|}^{r}\mathrm{sgn}\left({S}_{\psi }\right)+{u}_{d1}\\ \frac{\text{d}{u}_{d1}}{\text{d}t}={K}_{I1}\mathrm{sgn}\left({S}_{\psi }\right)\end{array}
Torque controller:
\left\{\begin{array}{l}{u}_{q}={K}_{P2}{|{S}_{{T}_{e}}|}^{r}\mathrm{sgn}\left({S}_{{T}_{e}}\right)+{u}_{q1}\\ \frac{\text{d}{u}_{q1}}{\text{d}t}={K}_{I2}\mathrm{sgn}\left({S}_{{T}_{e}}\right)\end{array}
{S}_{\psi }
is the difference between a given flux linkage and the actual flux linkage;
{S}_{{T}_{e}}
is the difference between the given torque and the actual torque;
{K}_{P1},{K}_{I1},{K}_{P2},{K}_{I2}
3.2. Synovial Accessibility and Stability Analysis
According to the stator flux vector reference system,
{\psi }_{s}={\psi }_{d}
, the stator flux linkage is derived [11] :
\frac{\text{d}{\psi }_{s}}{\text{d}t}={u}_{d}-{R}_{s}{i}_{d}
Secondary derivation of the flux linkage:
\frac{{\text{d}}^{2}{\psi }_{s}}{\text{d}{t}^{2}}=\frac{{R}_{s}^{2}}{{L}_{s}}{i}_{d}-p{R}_{s}\omega {i}_{q}-\frac{{R}_{s}}{{L}_{s}}{u}_{d}+{\stackrel{˙}{u}}_{d}
{R}_{s},{L}_{s},{i}_{d},p,\omega
are bounded values, and therefore the formula (14) satisfies the stable condition of the formula (9).
For the hidden pole permanent magnet synchronous motor, assuming that the stator flux linkage amplitude is constant, the torque is also derived:
\frac{\text{d}{T}_{e}}{\text{d}t}=\frac{3}{2}p{\psi }_{s}\frac{\text{d}{i}_{q}}{\text{d}t}
Continue to guide the torque:
\begin{array}{c}\frac{{\text{d}}^{2}{T}_{e}}{\text{d}{t}^{2}}=\frac{3}{2}p{\psi }_{f}\left[-\frac{{p}^{2}{\psi }_{f}}{J}{i}_{d}{i}_{q}+\frac{pB}{J}{i}_{d}\omega +\left(\frac{{R}_{s}^{2}}{{L}_{s}^{2}}-\frac{{p}^{2}{\psi }_{f}^{2}}{J{L}_{s}}-{p}^{2}{\omega }^{2}\right){i}_{q}+\frac{2p{R}_{s}}{{L}_{s}}\omega {i}_{d}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{p\omega }{{L}_{s}}{u}_{sd}+\left(\frac{p{\psi }_{f}B}{J{L}_{s}}+\frac{p{\psi }_{f}{R}_{s}}{{L}_{s}^{2}}\right)\frac{p{\psi }_{f}B}{J{L}_{s}}\omega -\frac{{R}_{s}}{{L}_{s}^{2}}{u}_{sq}+\frac{p{\psi }_{f}{T}_{L}}{J{L}_{s}}+\frac{{\stackrel{˙}{u}}_{sq}}{{L}_{s}}\right]\end{array}
It can be seen that the parameters of the second derivative of the torque are bounded values, and the analysis method is the same as the stator flux linkage, which also satisfies the stable condition of Equation (9).
3.3. Synovial Speed Controller Design
\frac{\text{d}{\omega }_{m}}{\text{d}t}=\frac{p}{J}\left({T}_{e}-{T}_{L}\right)
The design switching function is:
s={\omega }_{m}^{\ast }-{\omega }_{m}
{\omega }_{m}^{\ast }
is the given speed of the motor;
{\omega }_{m}
is the actual speed of the motor.
According to the concept of the approach law of Gao Weibing [7] , the design approach law is:
\stackrel{˙}{s}=-\epsilon {|s|}^{\alpha }\mathrm{sgn}\left(s\right)-ks
1>\alpha >0,\epsilon >0,k>0
Available from Equation (18):
\stackrel{˙}{s}=-{\stackrel{˙}{\omega }}_{m}=\frac{p}{J}\left({T}_{L}-{T}_{e}\right)=-\epsilon {|s|}^{\alpha }\mathrm{sgn}\left(s\right)-ks
Further, the slip film controller output expression is:
{T}_{e}=\frac{J}{p}\left[\epsilon {|s|}^{\alpha }\mathrm{sgn}\left(s\right)+ks\right]+{T}_{L}
Select the Lyapunov function as:
V=\frac{1}{2}{s}^{2}
Derivation of Equation (22) yields:
\stackrel{˙}{V}=\stackrel{˙}{s}s=s\left[-\epsilon {|s|}^{\alpha }\mathrm{sgn}\left(s\right)-ks\right]=-\left(\epsilon {|s|}^{\alpha +1}+k{s}^{2}\right)<0
This formula satisfies the stability conditions of Lyapulov, which proves the stability and accessibility of the system.
In this paper, the simulation model is built by matlab/simulink. The system includes multiple subsystem modules such as flux linkage and torque estimation module, coordinate transformation module and speed adjustment module. The improved system model is shown in Figure 1.
The parameters of permanent magnet synchronous motor are set as follows: the stator winding resistance is
{R}_{s}=1.2\text{\hspace{0.17em}}\Omega
, stator inductance
{L}_{d}={L}_{q}=0.0085\text{\hspace{0.17em}}\text{H}
, rotor flux linkage
{\psi }_{f}=0.175\text{\hspace{0.17em}}\text{Wb}
J=0.0008\text{\hspace{0.17em}}\text{kg}\cdot {\text{m}}^{2}
, pole logarithm p = 4, the system friction coefficient B = 0. The system simulation parameters are: given speed is
{\omega }_{m}^{\ast }=600\text{\hspace{0.17em}}\text{rad}/\text{min}
, given flux linkage is
{\psi }^{*}=0.3\text{\hspace{0.17em}}\text{Wb}
, load torque
{T}_{L}=1.5\text{\hspace{0.17em}}\text{N}\cdot \text{m}
is applied in 0.2 seconds. Simulation time is 0.4 s. The simulation results are shown in Figures 2-7.
Figure 1. Improved DTC simulation model for permanent magnet synchronous motor.
Figure 2. Traditional DTC motor speed curve.
Figure 3. Improve DTC motor speed curve.
Figure 4. Electromagnetic torque curve of traditional DTC motor.
Figure 5. Improve the electromagnetic torque curve of DTC motor.
Figure 6. Traditional DTC motor electromagnetic torque curve.
It can be seen from Figures 2-7 that when the motor speed increases from 0 to 600 r/min, compared with Figure 2 and Figure 3, the conventional direct torque control has a large overshoot and the adjustment time is long, when it is within 0.2 s. When the load is applied, the rotational speed drops significantly and the steady-state performance is poor. In the improved direct torque control, the rotational speed has a small overshoot and the dynamic response is fast, and the rotational speed is substantially unchanged when the load is suddenly applied. Comparing Figures 4-7, in the traditional direct torque control, the torque and flux linkage are large, and in the direct torque control using the sliding mode variable structure control, the torque and flux pulsation are significantly reduced, so it can be seen The method adopted in this paper is effective, can improve the system speed regulation performance, reduce the ripple of the flux linkage and torque, and make the system have good dynamic response and anti-interference.
In this paper, based on the sliding mode theory, the flux linkage and torque controller and the sliding mode variable speed controller are designed. Compared with traditional direct torque control simulation, the method used in this paper can effectively reduce the flux linkage and torque ripple. The system has strong anti-interference and robustness.
Cite this paper: Luo, Y. , Tan, G. and Su, C. (2019) Direct Torque Control of Permanent Magnet Synchronous Motor Based on Sliding Mode Variable Structure. Open Access Library Journal, 6, 1-10. doi: 10.4236/oalib.1105758.
[1] Luo, Z.W., Gu, A.Z. and Hong J.J. (2017) Research on Servo System of Permanent Magnet Synchronous Motor Based on Improved Sliding Mode Variable Structure Control. Machine Tool & Hydraulics, 19, 25-29.
[2] Jiang, C.F. (2008) Direct Torque Control System of Permanent Magnet Synchronous Motor Based on Double Fuzzy Control. Central South University, Changsha, 135-150.
[3] Jia, H.P. and He, Y.K. (2006) A Study on the Effect of Zero Vector in Direct Torque Control of Permanent Magnet Synchronous Motor. Electric Drive, 4, 23-26.
[4] Swierczynski, D. and Kazmierkowski, M.P. (2002) Direct Torque Control of Permanent Magnet Synchronous Motor (PMSM) Using Space Vector Modulation (DTC-SVM)-Simulation and Experimental Results. IEEE 2002 28th Annual Conference of the Industrial Electronics Society, Sevilla, Spain, 5-8 November 2002, 751-755. https://doi.org/10.1109/IECON.2002.1187601
[5] Wu, H., Mu, G. and Cui, Y.F. (2008) Direct Torque Fuzzy Control Technology of Permanent Magnet Synchronous Motor. Journal of Shenyang University of Technology, 30, 261-265.
[6] Feng, Y., Yu, X. and Han, F. (2013) High-Order Terminal Sliding-Mode Observer for Parameter Estimation of a Perma-nent-Magnet Synchronous Motor. IEEE Transactions on Industrial Electronics, 60, 4272-4280.
[7] Lai, Y.S., Wang, W.K. and Chen, Y.C. (2004) Novel Switching Techniques for Reducing the Speed Ripple of AC Drives with Direct Torque Control. IEEE Transactions on Industrial Electronics, 51, 768-775.
[8] Lu, G.Z., Hao, R.K. and Huang, J.H. (2018) Direct Torque Control of Permanent Magnet Synchronous Motor Based on Sliding Mode Variable Structure. Electronic Measurement Technology, (21)
[9] Ouyang, F. and Chen, L. (2018) New Approaching Sliding Mode Variable Structure Control of Permanent Magnet Synchronous Motor. Automation and Instrumenta-tion, 12, 21-25.
[10] Chen, F., Wang, M. and Liu, H.T. (2019) Research on PMSM DTC for Electric Vehicle Based on Improved Sliding Mode Variable Structure. Micro Motor, 1,79-82
[11] Wu, J., Dai, Y.H. and Tang, P. (2018) Research on Direct Torque Optimization Control of Permanent Magnet Synchronous Motor. Computer Simulation, 11, 341-346.
|
Why Mean Squared Error and L2 regularization? A probabilistic justification. – Avital Oliver
\ell_2
\ell_1
\ell_1
\ell_2
(This note is also available as a PDF.)
What is a regression problem? In simplest form, we have a dataset
\mathcal{D}=\{ (x_i \in \mathbb{R}^n, y_i \in \mathbb{R} ) \}
and want a function
f
that approximately maps
x_i
y_i
without overfitting. We typically choose a function (from some family
\Theta
) parametrized by
\theta
. A simple parametrization is
f_\theta:x \mapsto x \cdot \theta
\theta \in \Theta = \mathbb{R}^n
– this is linear regression. Neural networks are another kind of parametrization.
Now we use some optimization scheme to find a function in that family that minimizes some loss function on our data. Which loss function should we use? People commonly use mean squared error (aka
\ell_2
loss):
\frac{1}{|\mathcal{D}|}\sum(y_i - f_\theta(x_i))^2
Two assumptions: (1) Data is noisy; (2) We want the most likely model
The data is generated by a function in our family, parametrized by
\theta_\text{true}
, plus noise, which can be modeled by a zero-mean Gaussian random variable: \begin{equation} f_\text{data}(x) = f_{\theta_\text{true}}(x) + \epsilon \end{equation} \begin{equation} \epsilon \sim \mathcal{N}(0, \sigma^2) \end{equation} (Why Gaussian? We’ll get back to this question later.)
Given the data, we’d like to find the most probable model within our family. Formally, we’re looking for parameters
\theta
with the highest probability: \begin{equation} \operatorname*{arg\,max}_\theta(P(\theta \mid \mathcal{D})) \end{equation}
With these assumptions, we can derive
\ell_2
loss as the principled error metric to optimize. Let’s see how.
Probability of data given parameters
First, observe that with these two assumptions, we can derive the probability of a particular datapoint
(x, y)
\begin{align} P((x, y) \in \mathcal{D} \mid \theta) & = P(y=f_\theta(x) + \epsilon \mid \epsilon \sim \mathcal{N}(0, \sigma^2))) \\ & = \mathcal{N}(y - f_\theta(x); 0, \sigma^2) \\ & = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(y-f_\theta(x)^2}{2\sigma^2}} \end{align}
The math will be less complicated if we use log probability, so let’s switch to that here:
\begin{align} \log P((x, y) \in \mathcal{D} \mid \theta) & = \log \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(y-f_\theta(x)) ^2}{2\sigma^2}} \\ & = -\frac{(y-f_\theta(x)) ^2}{2\sigma^2} + const. \end{align}
Notice the
(y-f_\theta(x))^2
term above – that’s how we’re going to get the
\ell_2
loss. (Where did it come from? Could we have gotten something else there?)
Now we can extend this from the log probability of a data point to the log probability of the entire dataset. This requires us to assume that each data point is independently sampled, commonly called the i.i.d. assumption.
\begin{align} \log P(\mathcal{D} \mid \theta) & = \sum \log P(y_i=f_\theta(x_i) + \epsilon \mid \epsilon \sim \mathcal{N}(0, \sigma^2))) \\ & = -\frac{1}{2\sigma^2} \sum_{x, y \in \mathcal{D}} (y - f_\theta(x))^2 + const. \end{align}
That’s a simple formula for the probability of our data given our parameters. However, what we really want is to maximize the probability of the parameters given the data, i.e.
P(\theta \mid \mathcal{D})
Minimizing MSE is maximizing probability
We turn to Bayes’ rule,
P(\theta \mid \mathcal{D}) \propto P(\mathcal{D} \mid \theta) P(\theta)
, and find that:
\begin{align} \log P(\theta \mid \mathcal{D}) & = \log P(\mathcal{D} \mid \theta) + \log P(\theta) + const. \\ & = \left[ -\frac{1}{2\sigma^2} \sum_{x, y \in \mathcal{D}} (y - f_\theta(x))^2 \right] + \log P(\theta) + const. \end{align}
The term in the left-hand side logarithm,
P(\theta \mid \mathcal{D})
, is called the posterior distribution. The two non-constant right-hand side terms also have names:
P(\mathcal{D} \mid \theta)
is the likelihood, and
P(\theta)
is the prior distribution (the likelihood does not integrate to 1, so it’s not a distribution). The prior is a distribution we have to choose based on assumptions outside of our data. Let’s start with the simplest – the so-called uninformative prior
P(\theta) \propto 1
, which doesn’t describe a real probability distribution but still lets us compute the posterior. Choosing an uninformative prior corresponds to making no judgement about which parameters are more likely. If we choose the uninformative prior, we get:
\begin{align} \log P(\theta \mid \mathcal{D}) & = \log P(\mathcal{D} \mid \theta) + \log P(\theta) + const. \\ & = -\frac{1}{2\sigma^2} \sum_{x, y \in \mathcal{D}} (y - f_\theta(x))^2 + const. \end{align}
Ok woah. We’re there. Maximizing
P(\theta \mid \mathcal{D})
is the same as minimizing
\sum (y_i - f_\theta(x_i))^2
. The formal way of saying this is that minimizing mean squared error maximizes the likelihood of the parameters. In short, we’ve found the maximum likelihood estimator (MLE).
If we change our assumptions, though…
We can also change our assumptions and see what happens:
What if we change the variance on the noise? The log posterior which we’re maximizing changes by a constant factor, so the same model is most likely. We only needed to assume that the noise is drawn from some zero-mean Gaussian. (The variance matters if we place a prior as in (3) below)
If we assume a different type of noise distribution, we’d derive a different loss function. For example, if we model the noise as being drawn from a Laplace distribution, we’d end up with
\ell_1
error instead.
If we actually place a prior on our parameters we’d get a regularization term added to the log posterior that we’re maximizing. For example, if the prior is a zero-mean Gaussian, we’d get
\ell_2
regularization. And if the prior is a zero-mean Laplacian, we’d get
\ell_1
regularization. When we set a prior, we call the most likely parameters the maximum a posteriori estimate (MAP).
What if our models have different types of parameters, such as the layers in a neural network? We would still want to place a prior on them to avoid overfitting, but we’d want a different prior for different layers. This corresponds to choosing different regularization hyperparameters for each layer.
But don’t believe me – derive these yourself!
|
SAT Right Triangles | Brilliant Math & Science Wiki
To successfully solve problems about right triangles on the SAT, you need to know:
that the measures of the angles in a triangle add to
180^\circ
In a right triangle, the square of the hypotenuse equals the sum of the square of the legs.
b^2 = a^2 + c^2
how to determine if a triangle is right, acute, or obtuse, given the lengths of its sides
c^2 = a^2 + b^2,
m\angle C = 90
\triangle ABC
c^2 < a^2 + b^2,
m\angle C < 90
\triangle ABC
c^2 > a^2 + b^2,
m\angle C > 90
\triangle ABC
the relationship between the sides of the special right triangles
30^\circ-60^\circ-90^\circ
30^\circ-60^\circ-90^\circ
triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is
\sqrt{3}
times as long as the shorter leg.
45^\circ-45^\circ-90^\circ
45^\circ-45^\circ-90^\circ
triangle, the hypotenuse is
\sqrt{2}
times longer than either of the legs.
SAT Tips for Right Triangles
If the perimeter of a square is 16, which of the following is the length of its diagonal?
\ \ 2\sqrt{2}
\ \ 4
\ \ 4\sqrt{2}
\ \ 8\sqrt{2}
\ \ 16\sqrt{2}
Tip: Perimeter of a polygon equals the sum of the lengths of its sides.
a^2 + b^2 = c^2.
ABCD
has four sides of equal length. Its perimeter in terms of one of its sides,
\overline{AB}
\begin{aligned} 4\cdot AB &= P\\ 4\cdot AB &= 16\\ AB &= 4 \end{aligned}
This is shown in the picture below.
\triangle ABC
is a right triangle whose hypotenuse is the diagonal of the square. To find the length of the hypotenuse, we use the Pythagorean Theorem:
\begin{aligned} 4^2 + 4^2 &= AC^2\\ 16 + 16 &= AC^2\\ 32 &= AC^2\\ \sqrt{32} &= AC\\ 4\sqrt{2} &= AC\\ \end{aligned}
Note that since the length of a segment is a positive number, when square rooting we select the positive root.
If you calculate half the length of the diagonal, you will get this wrong answer.
This is the length of the side of the square, not the length of its diagonal.
If you calculate the sum of the lengths of both diagonals, you will get this wrong answer.
If you think the side of the square equals 16, you will get this wrong answer. The perimeter of the square is equal to 16, and one of its sides equals 4.
Which of the following CANNOT be the lengths of the sides of an obtuse triangle?
\ \ 5, 6, 10
\ \ 6, 6, 6\sqrt{2}
\ \ 6, 8, 11
\ \ 10, 15, 20
\ \ 20n, 30n, 40n
In the above figure, what is the length of
\overline{AB}
a?
\ \ a\sqrt{3}
\ \ 2a
\ \ a + \sqrt{3}
\ \ a + a\sqrt{2}
\ \ a+a\sqrt{3}
In the figure above, every triangle is a
30^\circ-60^\circ-90^\circ
triangle and, for each pair of triangles, the hypotenuse of the triangle to the right is twice as long as the long leg of the triangle to the left. If the short leg of the smallest triangle is
\sqrt{3},
a?
\ \ 3\sqrt{3}
\ \ 9
\ \ 8\sqrt{3}
\ \ 9\sqrt{3}
\ \ 18
In both solutions, we will refer to the smallest triangle as the 1st
\triangle
, and working our way clockwise, we will refer to the biggest triangle as the 4th
\triangle.
If you understand the relationship between the triangles and their sides, this problem can be done very quickly, like this:
a=\text{short leg of 1st}\ \triangle \cdot \sqrt{3} \cdot \sqrt{3} \cdot \sqrt{3} = \sqrt{3} \cdot \sqrt{3} \cdot \sqrt{3} \cdot \sqrt{3} = 9.
Solution 1 arrives at this result by applying the properties of
30^\circ - 60^\circ - 90^\circ
triangles for each triangle; Solution 2 explains it using similar triangles.
Tip: Know the
30^\circ-60^\circ-90^\circ
45^\circ-45^\circ-90^\circ
We are told that the short leg of the 1st
\triangle = \sqrt{3}.
30^\circ-60^\circ-90^\circ
triangle, the longer leg is
\sqrt{3}
times the shorter leg. Therefore,
\text{long leg of 1st}\ \triangle = \sqrt{3} \cdot \sqrt{3} = 3.
We are also given that the hypotenuse of each successive triangle is twice the long leg of the triangle with which it overlaps. So,
\text{hypotenuse of 2nd}\ \triangle = 2 \cdot \text{long leg of 1st}\ \triangle = 2 \cdot 3 = 6.
By the properties of
30^\circ-60^\circ-90^\circ
triangles,
\text{long leg of 2nd}\ \triangle = \frac{\sqrt{3}}{2} \cdot \text{hypotenuse of 2nd}\ \triangle = \frac{\sqrt{3}}{2} \cdot 6 = 3\sqrt{3}.
We apply the given again, this time to the 3rd
\triangle:
\text{hypotenuse of 3rd}\ \triangle = 2 \cdot \text{long leg of 2nd}\ \triangle = 2 \cdot 3\sqrt{3} = 6\sqrt{3}
and by the properties of
30^\circ-60^\circ-90^\circ
triangles:
\text{long leg of 3rd}\ \triangle = \frac{\sqrt{3}}{2} \text{hypotenuse of 3rd}\ \triangle = \frac{\sqrt{3}}{2} \cdot 6\sqrt{3} = 9.
We apply the given to the 4th
\triangle:
\text{hypotenuse of 4th}\ \triangle = 2 \cdot \text{long leg of 3rd}\ \triangle = 2 \cdot 9 = 18
and by the
30^\circ-60^\circ-90^\circ
a=\text{short leg of 1st}\ \triangle = \frac{\text{hypotenuse of 2nd}\ \triangle}{2} = \frac{18}{2} = 9.
30^\circ-60^\circ-90^\circ
45^\circ-45^\circ-90^\circ
Tip: AA Postulate: Two triangles are similar if two angles of one triangle are congruent to two angles of the other triangle.
\triangle = \sqrt{3}.
30^\circ-60^\circ-90^\circ
\sqrt{3}
\text{long leg of 1st}\ \triangle = \sqrt{3} \cdot \sqrt{3} = 3.
We are also told that the hypotenuse of each successive triangle is twice the long leg of the triangle with which it overlaps. So,
\text{hypotenuse of 2nd}\ \triangle = 2 \cdot \text{long leg of 1st}\ \triangle = 2 \cdot 3 = 6.
Applying the properties of
30^\circ-60^\circ-90^\circ
triangles to the 2nd
\triangle
\text{short leg of 2nd}\ \triangle = \frac{\text{hypotenuse of 2nd}\ \triangle}{2} = \frac{6}{2} = 3.
By the AA Similarity Postulate, the 2nd
\triangle
is similar to the 1st
\triangle,
and the factor of proportionality is
\frac{\text{short leg of 2nd}\ \triangle}{\text{short leg of 1st}\ \triangle} = \frac{3}{\sqrt{3}}.
In fact, by the AA Similarity Postulate, all four triangles are similar. And because the relationship between the sides of each successive pair of triangles is the same, the factor of proportionality between the 3rd and 2nd triangles, and between the 4th and 3rd triangles is also
\frac{3}{\sqrt{3}}.
\frac{\text{short leg of 3rd}\ \triangle}{\text{short leg of 2nd}\ \triangle} = \frac{3}{\sqrt{3}}
\text{short leg of 3rd}\ \triangle = \frac{3}{\sqrt{3}} \cdot \text{short leg of 2nd}\ \triangle
\text{short leg of 3rd}\ \triangle = \frac{3}{\sqrt{3}} \cdot 3 = \frac{9}{\sqrt{3}}
\frac{\text{short leg of 4th}\ \triangle}{\text{short leg of 3rd}\ \triangle} = \frac{3}{\sqrt{3}}
\text{short leg of 4th}\ \triangle = \frac{3}{\sqrt{3}} \cdot \text{short leg of 3rd}\ \triangle
a = \text{short leg of 4th}\ \triangle = \frac{3}{\sqrt{3}} \cdot \frac{9}{\sqrt{3}} = 9
If you miscount the number of triangles -- if you stop at the third triangle -- you will get this wrong answer.
Starting with the smallest triangle, if you think each successive short leg is twice as long as the short leg of the previous triangle, you will get this wrong answer.
If you count one too many triangles, you will get this wrong answer.
If you find the hypotenuse of the biggest triangle, instead of
a,
a^2 + b^2 = c^2.
c^2 = a^2 + b^2,
m\angle C = 90
\triangle ABC
c^2 < a^2 + b^2,
m\angle C < 90
\triangle ABC
c^2 > a^2 + b^2,
m\angle C > 90
\triangle ABC
30^\circ-60^\circ-90^\circ
45^\circ-45^\circ-90^\circ
180^\circ.
Cite as: SAT Right Triangles. Brilliant.org. Retrieved from https://brilliant.org/wiki/sat-right-triangles/
|
oliverpope | What are Algorithms?
An algorithm is something every person does every single day, it is the procedure or formula for solving a problem. For example, when an average person wakes up they will take a shower, do their hair, eat breakfast, drink coffee, watch the news, get dressed, and go to work. You could specify this as a person’s algorithm for getting ready for their day, it is simply the procedure one goes through to solve one’s problem.
In mathematics, an algorithm can simply be a function, such as
f(x)= x^2
. Where the procedure is finding where
y=0
and that would be
x
. Such as this equation would be 0, as shown below.
In computer science, an algorithm can be a lot harder to understand or interpret, however it is mostly the same thing. If, for instance, I wanted to know if 5 was greater than 1, I could use JavaScript to do it:
// simple if statement
// if 5>1 was an inequality, you could handle it with an else statement, however this is optional
console.log("this will never happen");
Algorithms can get much more complicated and hardcore than these simple examples, applications and programs are just elaborate algorithms that use special things like an if statement to make magic happen. All it is is common sense and logic, literally everyone does it.
|
How do you solve the system of equations x-y=7 and
How do you solve the system of equations
x-y=7
-2x+5y=-8
Your goal here is to remove one of the variables from the problem. You can see that the first equation has x and the second equation has -2x. If we double the first equation, we get:
2x-2y=14
Then we simply add that to the second equation:
2x-2y=14
±2x+5y=-8
3y=6
The positive 2x and the negative 2x cancel out, leaving us with just the
3y=6
Divide both sides by 3 and we get
y=2
Last, just plug 2 in for y in either equation (Ill
Begin by solving either the x or y variable by first eliminating or canceling out one of the variables. We can then plug in that variable into the first equation and solve for the second equation
x-y=7
-2x+5y=-8
To Solve for y in the second equation, start by multiplying the first equation by 2 and add the result to second in order to cancel out the
2\cdot \left(x-y=7\right)=2x-2y=14\to
Add this to the second equation
-2x+5y=-8
+2x-2y=14
3y=6
y=2
Now plug in the 2 for y in the first equation to solve for x
x-2+2=7+2
x=9
We have now solved both variables. Check to make sure both equations are equal.
Solve system of equations: These are linear equations. Since they are a system, both equations are solved simultaneously by substitution. The resulting values for x and y is the point at which the two lines intersect on a graph. The two equations are: x-y=7 and -2x+5y=-8 First Equation: x-y=7 x-y=7 Add y to both sides. x=7+y Second Equation: -2x+5y=-8 Substitute 7+yfor x in equation. -2(7+y)+5y=-8 Simplify. -14-2y+5y=-8 Add 14 to both sides. 2y+5y=-8+14 Simplify. 3y=6 Divide both sides by 3. y=2 Now substitute the value of y back into the firs equation and solve for x. x-2=7 Add 2 to both sides. x=9
A=\left[\begin{array}{cc}3& 1\\ 1& 1\\ 1& 4\end{array}\right],b\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]
\stackrel{―}{x}=
Use a triple integral to find the volume of the solid bounded by the graphs of the equations.
z=2-y,z=4-{y}^{2},x=0,x=3,y=0
\frac{dy}{dx}=\left(x+y-1\right)+\frac{x+y}{\mathrm{log}\left(x+y\right)}
Solve the equation 6x=6.
{\left(\frac{1}{4}\right)}^{x}=\frac{1}{64}
|
This is a quick start guide to streaming on Twitch as a 2D vtuber for people who have never streamed before.
Know how to install programs through Steam
Know how to do basic things like unzip files and install software
First you’ll need a Live2D model. If you don’t have one, you can use one of Live2D’s official sample models for testing. For this tutorial, I’ll use the Hiyori Momose model.
Here were the files inside after I extracted them.
Now you should install a face tracking application. There are a couple out there, but I currently recommend PrPrLive. Launch PrPrLive and hit the [ Live2D ] button, the one that looks like a person.
Then click the pink [ Load ] button.
Then navigate to the folder where you have your Live2D model. In that folder there should be a folder called runtime, and then a file that ends in “model3.json”. (Click the picture to see the bigger version.)
Now after you click [ Open ], an icon should appear with the name of your model.
Click that icon and your webcam should turn on and you will see your model on screen. By holding alt and using the scroll wheel, you can resize the model up and down. By holding alt and dragging, you can move your model.
If everything is working right, the model should copy your face movements.
Now I’ll show how to start streaming. Install OBS Studio but don’t launch it yet. Before launching it, we need to install the PrPrLive OBS Plugin. From inside PrPrLive, click the [ Video Streaming ] button. (If you launched it just now, just close it.)
Click [ Auto Install ] and acknowledge any dialogs that pop up.
Now launch OBS Studio. Click [ Next ].
Click [ Next ]. You can adjust these settings later.
Click [ Connect Account (recommended) ] and follow the on screen instructions to connect to your Twitch account. If you are using gmail and they email you a verification code, it might end up in your Promotions folder.
Click [ Apply Settings ]
You’ll be taken to a blank screen. I like to dock the stream-related windows to the left and right of the center window. You can do that by dragging those windows into their docking positions.
At the bottom, click the [ + ] in the Sources panel to add a source.
Select PrPrLive
\alpha
-channel. If you do not see this option, try redoing the PrPrLive OBS Plugin step and restarting OBS Studio.
Now click [ Start Streaming ] and you’ll start broadcasting live on Twitch!
← The Multivariate Gaussian For Programmers
Live2D Rigging Basics →
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Fluid Mechanics | Brilliant Math & Science Wiki
Andrew Ellinor, Agnibha Sen, Priyansh Sangule, and
Fluid mechanics is the physics of flowing matter, which includes, but is not limited to, cars moving through the traffic grid, waste flowing through the sewer system, gases moving through an engine, or sap moving sucrose from the leaves to the distal parts of a tree. Other examples of fluid mechanics include buoyancy (why you'll float in the Dead Sea), surface tension, wound healing, pattern formation in boiling liquids (the so-called Rayleigh-Bènard convection), and the motion of ants or flocks of birds moving in unison.
Lava, flowing molten rock, is an example of a fluid
Fluids have a bad reputation because their detailed behavior is governed by the Navier-Stokes equations, which pose great mathematical difficulties. Happily, it is often unnecessary to solve these equations to obtain great insight to the behavior of fluids. One can often make progress with the right set of approximations. To start, we'll analyze the mechanics of a very simple problem, that of floating.
If you jump into water, chances are you'll float, whether it be salt or freshwater, but if you throw an anvil, it goes straight to the bottom. The reason why one floats and the other sinks is very simple and can be obtained by a straightforward consideration of a pool of water, free of people, anvils, or other debris. Consider some selection of water,
\Gamma_S
within the pool, of volume
V_{\textrm{H}_2\textrm{O}}
of arbitrary shape, indicated by the black closed circle in the diagram below.
Because the pool is in a gravitational field,
\Gamma_S
is pulled down by the force
F=m_{\textrm{H}_2\textrm{O}}g = V_{\textrm{H}_2\textrm{O}}\rho_{\textrm{H}_2\textrm{O}}g
Now, the pool maintains a stable shape within the container, therefore the forces on any such arbitrary volume are balanced. Concretely, any volume
V_{\textrm{H}_2\textrm{O}}
in the pool experiences an upward force
V_{\textrm{H}_2\textrm{O}}\rho_{\textrm{H}_2\textrm{O}}g
, called the buoyant force.
The pool doesn't know that
\Gamma_S
is a volume of water, if we replaced
\Gamma_S
with the same volume of some other substance with the same weight as
\Gamma_S
, it would feel the same buoyant force and would float in place. In fact, if we replaced
\Gamma_S
with any object of the same volume, it would feel the same buoyant force, regardless of its weight. Hence, we have arrived at Archimede's principle.
The buoyant force on a body is vertical and is equal and opposite to the weight of the fluid displaced by the body.
F_\text{buoyant}=\rho_\text{fluid}V_\text{object}g
This idea holds whether the fluid is water, molasses, air, or some other simple fluid.
Archimedes' principle can explain why frozen ice cubes rise to the top of your drink. Upon freezing, water undergoes a volume expansion, so that a volume
V
of liquid water, ends up as a volume
V^\prime > V
of frozen ice. Because the force it feels is proportional to its large volume, while the pull of gravity is proportional to its unchanged weight, the ice feels a net force upward
F = \rho g\left(V^\prime - V\right)
Even for objects that ultimately sink, Archimede's principle suggests an apparent weight reduction. When walking in water, a human of mass
m
who usually feels like they weigh
mg
will feel a reduced weight of
mg-V_\text{human}\rho_{\textrm{H}_2\textrm{O}}g
. Because the density of flesh is approximately 0.97 g/mL, the weight of a typical human will be
\begin{aligned} &mg - \frac{m}{\rho_\text{flesh}}\rho_{\textrm{H}_2\textrm{O}}g \\ &= mg\left(1-\frac{\rho_{\textrm{H}_2\textrm{O}}}{\rho_\text{flesh}}\right)\\ &\approx 0.03\times mg \end{aligned}
just 3% of their normal weight. This makes swimming pools a convenient place for physical therapists who need to start teaching people to walk before their legs are strong enough to walk normally.
0.92 \text{ g/mL}
1.02 \text{ g/mL}
For fluids in active flow, we need something better than the balance of forces. Fortunately, we can get a long way with one simple assumption, the conservation of matter. When fluid moves from one position to the next, it must do so in such a way that no fluid matter is destroyed.
For instance, if we inject fluid into the mouth of a tube at a rate of
J_\text{in}=\SI{1}{\liter\per\second}
, we should find that
\SI{1}{\liter}
of the fluid comes out the other side every second, i.e. that
J_\text{in}=J_\text{out}
For a toy model, consider a the device below that consists of one level section of tube
\Gamma_\text{A}
with cross section of area
A_A
, connected by a linker section to another level tube
\Gamma_\text{B}
with cross section
A_B
We want to find a relationship that connects the velocity and pressure of the fluid in either section of tube.
To start, let us take our system to be the fluid that's between the discs
\partial_\text{A}
\partial_\text{B}
at time zero, which we call
\Sigma
. In order for
\Sigma
to flow to the right, there must be a net force to push it along. If the fluid pressure in
\Gamma_\text{A}
P_A
is greater than the fluid pressure in
\Gamma_{B}
P_B
, then the fluid to the left of
\Sigma
\Sigma_L
will push with greater force than the fluid to the right,
\Sigma_R
\Sigma
will flow. These two forces will be
P_A A_A
P_B A_B
Applying the work energy principle, we have
W = \Delta T + \Delta U
. For our parcel of fluid, work is performed by the two pressures in moving
\Sigma
along the tube. Consider the flow undertaken in some span of time
\Delta t
\Gamma_A
\Sigma
will move the distance
v_A\Delta t
\Gamma_B
\Sigma
v_B\Delta t
. Hence, the net work
W
on the fluid is given by the work done on our fluid by the fluid to the left
\Sigma_L
F dx = P_A A_A v_A \Delta t
, minus the work done by our fluid to the fluid on the right
\Sigma_R
P_B A_B v_B \Delta t
W = P_A A_A v_A \Delta t - P_B A_B v_B \Delta t
However, we have the conservation condition
J_\text{in} = J_\text{out}
which applies for the discs
\partial_A
\partial_B
. Hence, it must be true that
A_A v_A \Delta t = A_B v_A \Delta t
In other words, the volume of fluid
\Delta m
that flows through
\partial_A
is equal to the mass of fluid that flows out from
\partial_B
W = \Delta V \left(P_A-P_B\right)
Now, the work done on
\Sigma
is equal to the change in energy of the fluid. The kinetic energy of our fluid
\Sigma
should be the same as before with the exception that the quantity
\Delta m
of liquid which initially moved with velocity
v_A
\Gamma_A
is now traveling with velocity
v_B
\Gamma_B
\Delta T = \frac12 \rho\Delta V v_B^2 - \frac12 \rho\Delta V v_A^2
\Delta V \left(P_A - P_B\right) = \frac12 \rho \Delta V \left( v_B^2 - v_A^2 \right)
P_A + \frac12\rho v_A^2 = P_B + \frac12\rho v_B^2
, which is Bernoulli's relation for fluid flow in an arbitrary tube of level height.
In this derivation, the tubes were kept at equal level for simplicity's sake. It is trivial to recalculate our relation for the case when the two tube sections are of differing heights in a gravitational field, as occurs for the plumbing system in an apartment building. In this case, the work-energy principle is given by
W = \Delta T + \Delta U
and we have thus have the full Bernoulli relation
P_A + \frac12\rho v_A^2 + \rho gh_A = P_B + \frac12\rho v_B^2 + \rho gh_B
Note that our calculation did not depend in any way on the particular setup that we used (the two tubes and linker section). The same calculation applies to a tube of arbitrary shape which carries out an arbitrary trajectory through a gravitational field. Thus, the relation can be used to connect any two cross sections of a fluid's flow.
Bernoulli's relation has a number of applications, particularly in the use of hydraulics.
^2
^2
In this problem, the fluid is transmitting the force from the operators foot to the brake pad. Brake pads hover very close to the disc they press, so we can assume that the fluid velocity is negligible. Indeed, once the foot is clamped down on the lever, the arrangement will be static. In this case, Bernoulli's relation simplifies to
P_\text{foot} = P_\text{brake}
As pressure is given by the force per unit area, we have
\frac{F_\text{foot}}{A_\text{foot}} = \frac{F_\text{brake}}{A_\text{brake}}
F_\text{foot} = F_\text{brake} \times \frac{A_\text{foot}}{A_\text{brake}} 1.5 \text{N}
1~\mbox{atm}
1~\mbox{atm}=101,325~\mbox{Pa}
-9.8~\mbox{m/s}^2
1~\mbox{g/cm}^3
A lemonade vat is essentially a big cylinder that rests on its end with a spigot on the very bottom. One particular vat is a cylinder with radius
0.1~\mbox{m}
and a spigot of radius
0.01~\mbox{m}
. Initially the lemonade is at a height of
0.5~\mbox{m}
in the vat. You then open the spigot. How long does it take for all the lemonade to flow out of the vat in seconds?
The vat is open to the air at the top.
-9.8~\mbox{m/s}^2
Cite as: Fluid Mechanics. Brilliant.org. Retrieved from https://brilliant.org/wiki/fluid-mechanics/
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The EV/EBITDA multiple for a company can be found by comparing the enterprise value, or EV, to the earnings before interest, taxes, depreciation and amortization, or EBITDA.
The EV/EBITDA Multiple Ratio
The EV/EBITDA ratio is a metric widely used to help investors determine the value of a business. It compares a company’s value, including debt and liabilities, to the its true cash earnings, less noncash expenses. This metric is often used to compare values of companies that operate in the same industry. Lower values can be an indication a company has been undervalued. Generally, analysts interpret any EV/EBITDA value below 10 as positive; however, it is still important to consider the value in relation to EV/EBITDA values of similar firms.
The name of the ratio essentially gives away the formula used for its calculation:
\begin{aligned} &\text{Value} = \frac{ EV }{ EBITDA } \\ &\textbf{where:}\\ &EV = \text{Enterprise value} \\ &EBITDA = \text{Earnings before interest, taxes,} \\ &\text{depreciation and amortization} \\ \end{aligned}
Value=EBITDAEVwhere:EV=Enterprise valueEBITDA=Earnings before interest, taxes,depreciation and amortization
To determine the value, the company’s enterprise value is divided by its earnings before interest, taxes, depreciation, and amortization. Enterprise value is calculated as the company's total market capitalization plus debt and preferred shares, minus the company's total cash.
Benefits of the Metric
The EV/EBITDA multiple is often used in conjunction with, or instead of, the price-to-earnings, or P/E, ratio. The former is sometimes considered a better valuation tool for potential investors because it is not affected by changes in a company’s capital structure and makes it possible to obtain fair comparisons of companies that have different capital structures. One other advantage of the multiple is it eliminates the effects of noncash expenses that are not typically a major consideration of investors.
EBITDA/EV Multiple
The EBITDA/EV multiple is a financial ratio that measures a company's return on investment.
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Model Platform Motion Using Trajectory Objects - MATLAB & Simulink - MathWorks Nordic
waypointTrajectory Example for Aircraft Landing
geoTrajectory Example For Flight Trajectory
kinematicTrajectory Example For UAV Path
kinematicTrajectory Example For Spacecraft Trajectory
Radar Toolbox™ provides three System objects that you can use to model trajectories of platforms including ground vehicles, ships, aircraft, and spacecraft. You can choose between these trajectory objects based on the available trajectory information and the distance span of the trajectory.
waypointTrajectory (Sensor Fusion and Tracking Toolbox) — Defines a trajectory using a few waypoints in Cartesian coordinates that the trajectory must pass through. The trajectory assumes the reference frame is a fixed North-East-Down (NED) or East-North-Up (ENU) frame. Since the trajectory interpolation assumes that the gravitational acceleration expressed in the trajectory reference frame is constant, waypointTrajectory is typically used for a trajectory defined within an area that spans only tens or hundreds of kilometers.
geoTrajectory (Sensor Fusion and Tracking Toolbox) — Defines a trajectory using a few waypoints in geodetic coordinates (latitude, longitude, and altitude) that the trajectory must pass through. Since the waypoints are expressed in geodetic coordinates, geoTrajectory is typically used for a trajectory from hundreds to thousands of kilometers of distance span.
kinematicTrajectory (Sensor Fusion and Tracking Toolbox) — Defines a trajectory using kinematic properties, such as acceleration and angular acceleration, expressed in the platform body frame. You can use kinematicTrajectory to generate a trajectory of any distance span as long as the kinematic information of the trajectory is available. The object assumes a Cartesian coordinate reference frame.
The two waypoint-based trajectory objects (waypointTrajectory and geoTrajectory) can automatically calculate the linear velocity information of the platform, but you can also explicitly specify the linear velocity using the Velocity property or a combination of the Course, GroundSpeed, and ClimbRate properties.
The trajectory of a platform is composed of rotational motion and translational motion. By default, the two waypoint-based trajectory objects (waypointTrajectory and geoTrajectory) automatically generate the orientation of the platform at each waypoint by aligning the yaw angle with the path of the trajectory, but you can explicitly specify the orientation using the Orientation property. Alternately, you can use the AutoPitch and AutoBank properties to enable automatic pitch and roll angles, respectively. For kinematicTrajectory, you need to use the Orientation property and the angular velocity input to specify the rotational motion of the trajectory.
The waypointTrajectory System object defines a trajectory that smoothly passes through waypoints expressed in Cartesian coordinates. Generally, you can use waypointTrajectory to model vehicles travelling within hundreds of kilometers. These vehicles include automobiles, surface marine craft, and commercial aircraft (helicopters, planes, and quadcopters). You can choose the reference frame as a fixed ENU or NED frame using the ReferenceFrame property. For more details on how the object generates the trajectory, see the Algorithms (Sensor Fusion and Tracking Toolbox) section of waypointTrajectory.
Define the trajectory of a landing aircraft using a waypointTrajectory object.
waypoints = [-421 -384 2000;
47 -294 1600;
-285 293 600;
-1274 84 350;
-2328 101 150;
-3209 83 0];
timeOfArrival = [0; 16.71; 76.00; 121.8; 204.3; 280.31; 404.33; 624.6];
aircraftTraj = waypointTrajectory(waypoints,timeOfArrival);
Create a theaterPlot object to visualize the trajectory and the aircraft.
minCoords = min(waypoints);
maxCoords = max(waypoints);
tp = theaterPlot('XLimits',1.2*[minCoords(1) maxCoords(1)], ...
'YLimits',1.2*[minCoords(2) maxCoords(2)], ...
'ZLimits',1.2*[minCoords(3) maxCoords(3)]);
% Create a trajectory plotter and a platform plotter
tPlotter = trajectoryPlotter(tp,'DisplayName','Trajectory');
pPlotter = platformPlotter(tp,'DisplayName','Aircraft');
Obtain the Cartesian waypoints of the trajectory using the lookupPose object function.
sampleTimes = timeOfArrival(1):timeOfArrival(end);
numSteps = length(sampleTimes);
[positions,orientations] = lookupPose(aircraftTraj,sampleTimes);
plotTrajectory(tPlotter,{positions})
Plot the platform motion using an airplane mesh object.
mesh = scale(rotate(tracking.scenario.airplaneMesh,[0 0 180]),15); % Exaggerated scale for better visualization
view(20.545,-20.6978)
plotPlatform(pPlotter,positions(i,:),mesh,orientations(i))
% Uncomment the next line to slow the aircraft motion animation
% pause(1e-7)
In the animation, the yaw angle of the aircraft aligns with the trajectory by default.
Create a second aircraft trajectory with the same waypoints as the first aircraft trajectory, but set its AutoPitch and AutoBank properties to true. This generates a trajectory more representative of the possible aircraft maneuvers.
aircraftTraj2 = waypointTrajectory(waypoints,timeOfArrival, ...
Plot the second trajectory and observe the change in aircraft orientation.
[positions2,orientations2] = lookupPose(aircraftTraj2,sampleTimes);
plotPlatform(pPlotter,positions2(i,:),mesh,orientations2(i));
Visualize the orientation differences between the two trajectories in angles.
distance = dist(orientations2,orientations);
plot(sampleTimes,distance*180/pi)
ylabel('Angular Ditance (dge)')
title('Orientation Difference Between Two Waypoint Trajectories')
The geoTrajectory System object generates a trajectory using waypoints in a similar fashion as the waypointTrajectory object, but it has two major differences in how to specify waypoints and velocity inputs.
When specifying waypoints for geoTrajectory, express each waypoint in the geodetic coordinates of latitude, longitude, and altitude above the WG84 ellipsoid model. Using geodetic coordinates, you can conveniently specify long-range trajectories, such as airplane flight trajectory on a realistic Earth model.
When specifying the orientation and velocity information for each waypoint, the reference frame for orientation and velocity is the local NED or ENU frame defined under the current trajectory waypoint. For example, the N1-E1-D1 frame shown in the figure is a local NED reference frame.
Ex, Ey, and Ez are the three axes of the Earth-centered Earth-fixed (ECEF) frame, which is fixed on the Earth.
(λ1, ϕ1, h1) and (λ2, ϕ2, h2) are the geodetic coordinates of the plane at the two waypoints.
(N1, E1, D1) and (N2, E2, D2) are the two local NED frames corresponding to the two trajectory waypoints.
Bx, By, and Bz are the three axes of the platform body frame, which is fixed on the platform.
Load the flight data of a flight trajectory from Los Angeles to Boston. The data contains flight information including flight time, geodetic coordinates for each waypoint, course, and ground speed.
load flightData.mat
Create a geoTrajectory object based on the flight data.
planeTraj = geoTrajectory([latitudes longitudes heights],timeOfArrival, ...
'Course',courses,'GroundSpeed',speeds);
Look up the Cartesian coordinates of the waypoints in the ECEF frame.
positionsCart = lookupPose(planeTraj,sampleTimes,'ECEF');
Show the trajectory using the helperGlobeView class, which approximates the Earth sphere.
viewer = helperGlobeView;
plot3(positionsCart(:,1),positionsCart(:,2),positionsCart(:,3),'r')
You can further explore the trajectory by querying other outputs of the trajectory.
Unlike the two waypoint trajectory objects, the kinematicTrajectory System object uses kinematic attributes to specify a trajectory. Think of the trajectory as a numerical integration of the pose (position and orientation) and linear velocity of the platform, based on the linear acceleration and angular acceleration information. The pose and linear velocity are specified with respected to a chosen, fixed scenario frame, whereas the linear acceleration and angular velocity are specified with respected to the platform body frame.
Create a kinematicTrajectory object for simulating a UAV path. Specify the time span of the trajectory as 120 seconds.
traj = kinematicTrajectory('SampleRate',1, ...
tspan = tStart:tFinal;
positions = NaN(3,numSteps);
velocities = NaN(3,numSteps);
vel = NaN(1,numSteps);
To form a square path covering a small region, separate the UAV trajectory into six segments:
Taking off and ascending in the z-direction
Moving in the positive x-direction
Moving in the positive y-direction
Moving in the negative x-direction
Moving in the negative y-direction
Descending in the z-direction and landing
In each segment, the UAV accelerates in one direction and then decelerates in that direction with the same acceleration magnitude. As a result, at the end of each segment, the velocity of the UAV is zero.
segSteps = floor(numSteps/12);
accelerations = zeros(3,numSteps);
% Acceleration for taking off and ascending in the z-direction
accelerations(3,1:segSteps) = acc;
accelerations(3,segSteps+1:2*segSteps) = -acc;
% Acceleration for moving in the positive x-direction
accelerations(1,2*segSteps+1:3*segSteps) = acc;
accelerations(1,3*segSteps+1:4*segSteps) = -acc;
% Acceleration for moving in the positive y-direction
% Acceleration for moving in the negative x-direction
% Acceleration for moving in the negative y-direction
accelerations(2,9*segSteps+1:10*segSteps) = acc;
% Descending in the z-direction and landing
accelerations(3,10*segSteps+1:11*segSteps) =-acc;
accelerations(3,11*segSteps+1:end) = acc;
Simulate the trajectory by calling the kinematicTrajectory object with the specified acceleration.
[positions(:,i),~,velocities(:,i)] = traj(accelerations(:,i)');
vel(i) = norm(velocities(:,i));
Visualize the trajectory using theaterPlot.
% Create a theaterPlot object and create plotters for the trajectory and the
% UAV Platform.
tp = theaterPlot('XLimits',[-30 130],'YLimits',[-30 130],'ZLimits',[-30 130]);
tPlotter = trajectoryPlotter(tp,'DisplayName','UAV trajectory');
pPlotter = platformPlotter(tp,'DisplayName','UAV','MarkerFaceColor','g');
plotTrajectory(tPlotter,{positions'})
view(-43.18,56.49)
% Use a cube to represent the UAV platform.
dims = struct('Length',10, ...
% Animate the UAV platform position.
plotPlatform(pPlotter,positions(:,i)',dims,eye(3))
Show the velocity magnitude of the UAV platform.
plot(tspan,vel)
title('Magnitude of the UAV Velocity')
Use kinematicTrajetory to specify a circular spacecraft trajectory. The orbit has these elements:
Orbital radius (
\mathit{r}
) — 7000 km
Inclination (
\mathit{i}
) — 60 degrees
Argument of ascending node (
\Omega
) — 90 degrees. The ascending node direction is aligned with the
\mathit{Y}
True anomaly (
\nu
\mathit{X}
\mathit{Y}
\mathit{Z}
is the Earth-centered inertial (ECI) frame, which has a fixed position and orientation in space.
\mathit{x}
\mathit{y}
\mathit{z}
is the spacecraft body frame, fixed on the spacecraft.
\stackrel{\to }{\mathit{r}}
\stackrel{\to }{\mathit{v}}
are the initial position and velocity of the spacecraft.
To specify the circular orbit using kinematicTrajectory, you need to provide the initial position, initial velocity, and initial orientation of the spacecraft with respect to the ECI frame. For the chosen true anomaly (
\mathit{v}\text{\hspace{0.17em}}
{90}^{\circ }
), the spacecraft velocity is aligned with the
\mathit{Y}
inclination = 60; % degrees
mu = 3.986e14; % standard earth gravitational parameter
radius = 7000e3;% meters
v = sqrt(mu/radius); % speed
initialPosition = [radius*cosd(inclination),0,-radius*sind(inclination)]';
initialVelocity = [0 v 0]';
Assume the x-direction of the spacecraft body frame is the radial direction, the z-direction is the normal direction of the orbital plane, and the y-direction completes the right-hand rule. Use the assumptions to specify the orientation of the body frame at the initial position.
orientation = quaternion([0 inclination 0],'eulerd','zyx','frame');
Express the angular velocity and the angular acceleration of the trajectory in the platform body frame.
omega = v/radius;
angularVelocity = [0 0 omega]';
a = v^2/radius;
acceleration = [-a 0 0]';
Set a simulation time of one orbital period. Specify a simulation step as 2 seconds.
tFinal = 2*pi/omega;
sampleRate = 1/dt;
tspan = 0:dt:tFinal;
Create the spacecraft trajectory. Since the acceleration and angular velocity of the spacecraft remain constant with respect to the spacecraft body frame, specify them as constants. Generate position and orientation outputs along the trajectory by using the kinematicTrajectory object.
traj = kinematicTrajectory('SampleRate',sampleRate, ...
'Position',initialPosition, ...
'Velocity',initialVelocity, ...
'AngularVelocity',angularVelocity, ...
'Acceleration',acceleration, ...
% Generate position and orientation outputs.
orientations = zeros(numSteps,1,'quaternion');
[positions(:,i),orientations(i)] = traj();
Use the helperGlobeView class and theaterPlot to show the trajectory.
viewer = helperGlobeView(0,[60 0]);
tp = theaterPlot('Parent',gca,...
'XLimits',1.2*[-radius radius],...
'YLimits',1.2*[-radius radius],...
'ZLimits',1.2*[-radius radius]);
tPlotter = trajectoryPlotter(tp,'LineWidth',2);
pPlotter = platformPlotter(tp,'MarkerFaceColor','m');
legend(gca,'off')
% Use a cube with exaggerated dimensions to represent the spacecraft.
dims = struct('Length',8e5,'Width',4e5,'Height',2e5,'OriginOffset',[0 0 0]);
plotPlatform(pPlotter,positions(:,i)',dims,orientations(i))
% Since the reference frame is the ECI frame, earth rotates with respect to it.
rotate(viewer,dt)
In this topic, you learned how to use three trajectory objects to customize your own trajectories based on the available information. In addition, you learned the fundamental differences in applying them. This table highlights the main attributes of these trajectory objects.
Linear Velocity Inputs
Acceleration and Angular Velocity Inputs
Recommended Distance Span
waypointTrajectory Cartesian waypoints expressed in a fixed frame (NED or ENU)
Automatically generate velocity for a smooth trajectory, by default
Specify velocity in the fixed frame at each waypoint
Specify course, ground speed, and climb rate in the fixed frame at each waypoint
Auto yaw by default, auto pitch by selection, and auto bank by selection
Specify orientation in the fixed frame
Cannot specify From within tens to hundreds of kilometers
geoTrajectory Geodetic waypoints in the ECEF frame
One of the these options:
Specify velocity in the local frame (NED or ENU) for each waypoint
Specify course, ground speed, and climb rate in the local frame (NED or ENU) for each waypoint
Specify orientation in the local frame
Cannot specify From hundreds to thousands of kilometers
kinematicTrajectory Initial position expressed in a chosen, fixed frame Only initial velocity in the fixed frame Only initial orientation in the fixed frame Specify acceleration and angular velocity in the platform body frame Unlimited distance span
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Solve B please: (please take D on BC,there's a printing error) - Maths - Congruency of Triangles - 16881917 | Meritnation.com
Solve B please: (please take D on BC,there's a printing error)
Solution) As, the corresponding sides of similar triangles
∆ADB~∆CDA\left(already proved in A part\right)
are proportional.
\frac{CD}{AD}=\frac{DA}{DB}\phantom{\rule{0ex}{0ex}}A{D}^{2}=CD×DB=18×8=144\phantom{\rule{0ex}{0ex}}AD=12cm\phantom{\rule{0ex}{0ex}}Regards!
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How do you solve for y: 2x-3y=6 ?
How do you solve for y:
2x-3y=6
Let us proceed to solve the equation 2x - 3y = 6 We take the term 3y to the right-hand side of the equation which gives 2x = 6 + 3y Now we bring 6 to the left side of the equation 2x - 6 = 3y And lastly, we bring the 3 also to the other side of the equation thus dividing the term (2x-6) by 3 which ultimately gives us the value of y. (2x-6)/3=y
Substract 2x from both sides of the equation.
-3y=6-2x
Divide each term by -3 and simplify.
Divide each term in -3y=6-2x by -3
\frac{-3y}{3}=\frac{6}{-3}+\frac{-2x}{-3}
Cancel the common factor of -3
y=\frac{6}{-3}+\frac{-2x}{-3}
y=-2+\frac{2x}{3}
You must do the same action on both sides of the = sign;
2x=6+3y
2x-6=3y
\frac{2x-6}{3}=y
y=\left(\frac{2}{3}\right)x-2
A=\left[\begin{array}{cc}3& 1\\ 1& 1\\ 1& 4\end{array}\right],b\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]
\stackrel{―}{x}=
2.(1,2)
\left\{\begin{array}{c}3x-y=1\\ 2x+3y=8\end{array}\right\}
A system of linear equations is given below.
2x+4y=10
-\frac{1}{2}x+3=y
C. There are infinite solutions.
D. There are no solutions.
Use the discriminant to determine whether each quadratic equation has two unequal real solutions, a repeated real solution (a double root), or no real solution, without solving the equation.
25{x}^{2}-20x+4=0
3\left(x-2\right)-2\left(3x+5\right)=4\left(x-1\right)
\frac{3+5x}{5}=\frac{4-x}{7}
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End-to-End DVB-S2X Simulation with RF Impairments and Corrections for Regular Frames - MATLAB & Simulink - MathWorks í•œêµ
{\mathit{E}}_{\mathit{s}}/{\mathit{N}}_{\mathit{o}}
{\mathit{E}}_{\mathit{s}}/{\mathit{N}}_{\mathit{o}}
{\mathit{E}}_{\mathit{s}}/{\mathit{N}}_{\mathit{o}}
{\mathit{E}}_{\mathit{s}}/{\mathit{N}}_{\mathit{o}}
{\mathit{E}}_{\mathit{s}}/{\mathit{N}}_{\mathit{o}}
{\mathit{E}}_{\mathit{s}}/{\mathit{N}}_{\mathit{o}}
{\mathit{E}}_{\mathit{s}}/{\mathit{N}}_{\mathit{o}}
{\mathit{E}}_{\mathit{s}}/{\mathit{N}}_{\mathit{o}}
{\mathit{E}}_{\mathit{s}}/{\mathit{N}}_{\mathit{o}}
E. Casini, R. De Gaudenzi, and Alberto Ginesi. "DVBâ€S2 modem algorithms design and performance over typical satellite channels." International Journal of Satellite Communications and Networking 22, no. 3 (2004): 281-318.
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Compute polynomial coefficients that best fit input data in least-squares sense - Simulink - MathWorks India
Least Squares Polynomial Fit
Compute polynomial coefficients that best fit input data in least-squares sense
Math Functions / Polynomial Functions
dsppolyfun
The Least Squares Polynomial Fit block computes the coefficients of the nth order polynomial that best fits the input data in the least-squares sense, where you specify n in the Polynomial order parameter. A distinct set of n+1 coefficients is computed for each column of the M-by-N input, u.
For a given input column, the block computes the set of coefficients, c1, c2, ..., cn+1, that minimizes the quantity
\sum _{i=1}^{M}{\left({u}_{i}-{\stackrel{^}{u}}_{i}\right)}^{2}
where ui is the ith element in the input column, and
{\stackrel{^}{u}}_{i}=f\left({x}_{i}\right)={c}_{1}{x}_{i}^{n}+{c}_{2}{x}_{i}^{n-1}+\text{ }...\text{ }+{c}_{n+1}
The values of the independent variable, x1, x2, ..., xM, are specified as a length-M vector by the Control points parameter. The same M control points are used for all N polynomial fits, and can be equally or unequally spaced. The equivalent MATLAB® code is shown below.
c = polyfit(x,u,n) % Equivalent MATLAB code
For convenience, the block treats length-M unoriented vector input as an M-by-1 matrix.
Each column of the (n+1)-by-N output matrix, c, represents a set of n+1 coefficients describing the best-fit polynomial for the corresponding column of the input. The coefficients in each column are arranged in order of descending exponents, c1, c2, ..., cn+1.
In the ex_leastsquarespolyfit_ref model below, the Polynomial Evaluation block uses the second-order polynomial
y=-2{u}^{2}+3
to generate four values of dependent variable y from four values of independent variable u, received at the top port. The polynomial coefficients are supplied in the vector [-2 0 3] at the bottom port. Note that the coefficient of the first-order term is zero.
The Control points parameter of the Least Squares Polynomial Fit block is configured with the same four values of independent variable u that are used as input to the Polynomial Evaluation block, [1 2 3 4]. The Least Squares Polynomial Fit block uses these values together with the input values of dependent variable y to reconstruct the original polynomial coefficients.
The values of the independent variable to which the data in each input column correspond. For an M-by-N input, this parameter must be a length-M vector. Tunable (Simulink).
The order, n, of the polynomial to be used in constructing the best fit. The number of coefficients is n+1.
Detrend DSP System Toolbox
Polynomial Evaluation DSP System Toolbox
Polynomial Stability Test DSP System Toolbox
polyfit MATLAB
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Extrapontine Myelinolysis-Induced Parkinsonism in a Patient with Adrenal Crisis
AbstractBackgroundDiscussionReferencesCopyrightRelated articles
Yahia Z. Imam, Maher Saqqur, Hassan Alhail, Dirk Deleu, "Extrapontine Myelinolysis-Induced Parkinsonism in a Patient with Adrenal Crisis", Case Reports in Neurological Medicine, vol. 2012, Article ID 327058, 3 pages, 2012. https://doi.org/10.1155/2012/327058
Yahia Z. Imam,1 Maher Saqqur,2 Hassan Alhail,1 and Dirk Deleu 1
1Neurology Section, Department of Medicine, Hamad Medical Corporation, P.O. Box 3050, Doha, Qatar
2Division of Neurology, University of Alberta, Edmonton, AB, Canada
Academic Editor: D. B. Fee
Background. Extrapontine myelinolysis (EPM) has been well described in the presence of rapid correction of hyponatremia. It is seldom reported with adrenal insufficiency. We report a unique case where a patient developed EPM as a result of adrenal insufficiency where the brain MRI revealed symmetrical lesion in the basal ganglia with pallidal sparing. Case Report. A 30-year-old gentleman with panhypopituitarism developed adrenal crisis, hyponatremia, and hyponatremic encephalopathy. Seven days after the rapid correction of hyponatremia, he developed parkinsonism and neuropsychiatric symptoms. MRI showed extrapontine myelinolysis without central pontine myelinolysis. Conclusion. Extrapontine myelinolysis without central pontine myelinolysis is rare and should raise a concern of associated adrenal insufficiency in the right clinical setting. Rapid correction of hyponatremia particularly in steroid-deficient states should be avoided as it can predispose to extrapontine myelinolysis. Magnetic resonance imaging is very helpful in supporting the diagnosis of EPM.
Central pontine myelinolysis (CPM) was first recognized in 1959 by Adams et al. [1]. In this paper; autopsy findings of myelin sheath destruction in a symmetrical fashion in the centre of the basis pontis were described. These lesions tend to spare the axons, the neuronal cell bodies, and the blood vessels with no signs of inflammation in the surrounding tissue. Malnutrition and alcohol consumption were the deemed causatives. Later on, the association was made with rapid correction of hyponatremia [2, 3]. Additionally liver disease, burns, and postliver transplantation were considered notorious culprits [2, 3].
It is now recognized that identical pathological demyelination to the ones seen in CPM can occur elsewhere, that is, extrapontine myelinolysis (EPM) either in combination with CPM or alone; collectively they were called osmotic demyelination [2, 3]. However, isolated EPM is relatively rare [2].
We presented a unique case of isolated EPM where a patient developed parkinsonism and neuropsychiatric symptoms 1 weeks after correction of hyponatremia in the setting of adrenal insufficiency.
A 30-year-old man known to have panhypopituitarism on replacement therapy suffered an adrenal crisis characterized by fever, abdominal pain, and vomiting, following a tooth extraction. This resulted in a severe hyponatremia of 105 mmol/L and hyponatremic encephalopathy manifesting as confusion, agitation, and stupor for which he was admitted. Sepsis workup was negative. Magnetic resonance imaging (MRI) and cerebrospinal fluid exams were normal. Electroencephalogram (EEG) showed only diffuse bilateral slowing. Thyroid function tests were within normal limits and random cortisol level as well as a synacthen test confirmed the diagnosis of adrenal insufficiency.
He was infused with normal saline to correct the hyponatremia as well as stress doses of hydrocortisone. After 72 hours his serum sodium level was 142 mmol/L (Figure 1). The patient’s general condition improved over the next 2-3 days. On day 9 after admission, he began to deteriorate again with the development of slowness of speech and movement, emotional liability, and swallowing difficulties progressing to severe hypomimia, rigidity in the upper limbs, and spasticity in the lower limb.
Serum Na by time.
MRI revealed EPM without CPM (Figures 2(a) and 2(b)) affecting symmetrically the basal ganglia and thalami but sparing the globus pallidi. In addition, there was increased signal intensity in both hippocampal regions.
(a) MRI brain: axial FLAIR image showing hyperintense signal in the caudate nuclei bilaterally (asterisk marks), both putamina bilaterally (white thick arrow pointing to the right putamen), and both thalami (thin white arrows). Note the sparring of the globus pallidi (black thick arrow). (b) Axial FLAIR of the brain showing increased signal in the temporal lobes (namely, the anterior temporal lobes, particularly the hippocampi (black arrow)). The Pons show no sign of demyelination.
To improve his parkinsonian syndrome he was empirically started on levodopa/carbidopa (125 mg tid) titrated up to control symptoms.
Followup after 2 months showed moderate improvement. The patient regained most of the activities of daily living after being totally dependent. He was able to ambulate without assistance and his parkinsonian symptoms were under control with the help of medicine.
In general parkinsonism, pseudobulbar symptoms, tetraparesis, and various movement disorders have been described with EPM [2, 4–9].
Our case report is unique for the following reasons. Firstly EPM without CPM is rare. Secondly, only 5 cases of EPM without CPM in association with adrenal insufficiency (Table 1) have been reported [4–9] and this is usually in the context of rapid correction of hyponatremia.
Article Cause of hyponatremia Symptomatology Outcome
Gujjar et al., 2010 [4] Addison’s disease and military tuberculosis (TB) Parkinsonism Good recovery
Al-Mamari et al., 2009 [5] Addison’s disease and miliary TB Parkinsonism Partial recovery
Srimanee et al.,
2009 [6] Hypopituitarism and secondary adrenal insufficiency Dystonia Not stated
Okada et al., 2005 [7] Hypopituitarism and secondary adrenal insufficiency Parkinsonism Good recovery
Lasheen et al.,
2005 [8] Panhypopituitarism, pituitary microadenoma, and secondary adrenal insufficiency Neuropsychiatric, dysarthria, and dystonia Not stated
Cases in the literature with parkinsonism and isolated EPM with adrenal insufficiency.
Typical MRI features of EPM include involvement of the cerebellum, the cerebral white matter, the basal ganglia (the most common site), and thalami, with sparing of the palladium [2, 9]. The lesions appear hyperintense on T2-weighted and FLAIR sequences and appear hypointense on T1 sequence [10, 11]. These findings alone are not specific for osmotic demyelination and must be interpreted in the appropriate clinical setting.
Treatment is usually symptomatically aimed at controlling parkinsonism, spasticity, and movement disorders and is often rewarding [4, 9].
Osmotic demyelination was originally regarded as carrying a grave prognosis with outcomes including death and severe disability. However, favourable outcomes are increasingly reported [9, 12, 13]. This is explained by the advancement of MRI in picking up the disease earlier in the course of the disease including the asymptomatic cases and the development of critical care services. However, MRI does not seem to predict prognosis or clinical improvement. The latter usually precedes radiological resolution if any [13].
In conclusion, EPM without CPM is rare and should raise a concern of associated adrenal insufficiency in the right clinical setting. Rapid correction of hyponatremia particularly in steroid deficient states should be avoided as it can predispose to EPM. A favourable prognosis is increasingly recognized and symptomatic treatment is the mainstay of management. MRI of the brain is very helpful in the diagnosis, but not so in terms of prognosis.
R. D. Adams, M. Victor, and E. L. Mancall, “Central pontine myelinolysis: a hitherto undescribed disease occurring in alcoholic and malnourished patients,” Archives of Neurology and Psychiatry, vol. 81, no. 2, pp. 154–172, 1959. View at: Google Scholar
R. J. Martin, “Central pontine and extrapontine myelinolysis: the osmotic demyelination syndromes,” Journal of Neurology in Practice, vol. 75, no. 3, pp. iii22–iii28, 2004. View at: Publisher Site | Google Scholar
D. G. Wright, R. Laureno, and M. Victor, “Pontine and extrapontine myelinolysis,” Brain, vol. 102, no. 2, pp. 361–385, 1979. View at: Google Scholar
A. Gujjar, A. Al-Mamari, P. C. Jacob, R. Jain, A. Balkhair, and A. Al-Asmi, “Extrapontine myelinolysis as presenting manifestation of adrenal failure: a case report,” Journal of the Neurological Sciences, vol. 290, no. 1-2, pp. 169–171, 2010. View at: Publisher Site | Google Scholar
A. Al-Mamari, A. Balkhair, A. Gujjar et al., “A case of disseminated tuberculosis with adrenal insufficiency,” Sultan Qaboos University Medical Journal, vol. 9, article 32, 2009. View at: Google Scholar
K. Okada, M. Nomura, N. Furusyo, S. Otaguro, S. Nabeshima, and J. Hayashi, “Amelioration of extrapontine myelinolysis and reversible Parkinsonism in a patient with asymptomatic hypopituitarism,” Internal Medicine, vol. 44, no. 7, pp. 739–742, 2005. View at: Publisher Site | Google Scholar
I. Lasheen, S. A. R. Doi, and K. A. S. Al-Shoumer, “Glucocorticoid replacement in panhypopituitarism complicated by myelinolysis a case report,” Medical Principles and Practice, vol. 14, no. 2, pp. 115–117, 2005. View at: Publisher Site | Google Scholar
J. Sajith, A. Ditchfield, and H. A. Katifi, “Extrapontine myelinolysis presenting as acute parkinsonism,” BMC Neurology, vol. 6, article 33, 2006. View at: Publisher Site | Google Scholar
G. M. Miller, H. L. Baker Jr., H. Okazaki, and J. P. Whisnant, “Central pontine myelinolysis and its imitators: MR findings,” Radiology, vol. 168, no. 3, pp. 795–802, 1988. View at: Google Scholar
P. Sharma, M. Eesa, and J. N. Scott, “Toxic and acquired metabolic encephalopathies: MRI appearance,” American Journal of Roentgenology, vol. 193, no. 3, pp. 879–886, 2009. View at: Publisher Site | Google Scholar
M. Koenig, J. P. Camdessanché, S. Duband, S. Charmion, J. C. Antoine, and P. Cathébras, “Extrapontine myelinolysis of favorable outcome in a patient with autoimmune polyglandular syndrome,” Revue de Medecine Interne, vol. 26, no. 1, pp. 65–68, 2005. View at: Publisher Site | Google Scholar
H. Menger and J. Jörg, “Outcome of central pontine and extrapontine myelinolysis (
n=44
),” Journal of Neurology, vol. 246, no. 8, pp. 700–705, 1999. View at: Publisher Site | Google Scholar
Copyright © 2012 Yahia Z. Imam 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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Circles - Intersecting Chords Practice Problems Online | Brilliant
r
be the radius of the above circle centered at
O.
\begin{array}{c}\displaystyle \lvert{\overline{PD}}\rvert=10, & \lvert{\overline{PC}}\rvert=15\end{array}
P
\overline{OB},
r^2?
The lengths of some line segments in the figure above are
\begin{array}{c}\displaystyle \lvert{\overline{AP}}\rvert=9, & \lvert{\overline{BP}}\rvert=6, & \lvert{\overline{OD}}\rvert=9.\end{array}
\lvert{\overline{OP}}\rvert=x,
x^2 ?
\begin{array}{c}\displaystyle \lvert{\overline{AP}}\rvert=13, & \lvert{\overline{CP}}\rvert=17, & \lvert{\overline{DP}}\rvert=8.\end{array}
\lvert{\overline{BP}}\rvert ?
{20}
\frac{136}{13}
\frac{272}{13}
{10}
If the lengths of some line segments in the figure above are
\begin{array}{c}\displaystyle \lvert{\overline{AP}}\rvert=6, & \lvert{\overline{OB}}\rvert=8, & \lvert{\overline{CP}}\rvert=x,\end{array}
x^2 ?
In the above circle centered at
O,
\lvert{\overline{OD}}\rvert=16,
\lvert{\overline{CD}}\rvert=8
\overline{OD} \perp \overline{AB}.
\lvert{\overline{CA}}\rvert^2 ?
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Accessing non-local data on Stack
In this article, we take a look at how a procedure accesses data used within it but doesn't belong to it.
Data access without nested procedures.
Issues with nested procedures.
Nested procedure declarations
Manipulation of access links.
Access links for procedure parameters.
In this article we shall first take a look at how access of non-local on the stack is done in the C programming language then transition into ML functional programming language since it permits nested functions and function declarations as first-class objects whereby a function can take another function as an argument and return a function as its value.
This is possible by the modifying the implementation of the run-time stack as we shall come to see.
In C languages, variables are defined either within a single function(local variables) or outside the function(global), on the other hand it is impossible to declare a procedure whose scope is within another procedure.
A global variable v has a scope consisting of all functions that follow the declaration of v except where there is a local definition of v's identifier.
Variables declared within a function will have a scope consisting of that function or part of it if the function has a nested block.
Languages which do not allow nested procedure declarations allocate storage variables and accesses them in the following way;
Global variables are allocated static storage meaning that the locations of these variables will be fixed and are known at compile time, therefore to access global variables to the executing procedure, we use a statically determined address.
Names are local to the activation at the top of the stack and may be accessed through the top_sp(from the prerequisite article) pointer of the stack.
A advantage of static allocation for global variables is that declared procedures are passed as parameters or returned as results without changing the data access strategy. e.g In C we pass a pointer to the function.
In C's static-scoping rule and without nested procedures, any name that is non-local to a procedure is non-local to all procedures regardless of their activation and if a procedure is returned as a result, any non-local name will reference its statically allocated storage location.
The static-scoping rule states that a procedure is permitted to access variables of the procedure whose declarations surround its own declaration.
Accessing non-local data is a complex task for languages which allow nesting of procedure declarations while using the static-scoping rule.
This is because knowing at compile time that a function p's declaration is immediately nested within a function q won't tell us the positions of their activation records at run-time since when either of the procedures are recursive they may have several activation records.
A static decision finds the declaration which applies to a non-local name x in a nested procedure p, it is made by extending the static-scope rule for blocks.
A dynamic decision finds q's activation from p's activation if x is declared in the enclosing procedure q.
The problem is that this requires additional run-time information about activations, we solve this using access links as we shall come to see.
Nested procedure declarations.
We introduce the ML programming language since it allows nested procedures.
Some properties of ML
It is a functional language and thus variables once declared and initialized are not changed except in cases e.g an array whose elements are changed by other functions.
It supports higher order functions, that is, a function can take functions as arguments, construct and return other functions which can in turn take functions as arguments to any level.
There are no iterations rather recursions. In ML functions call themselves with progressively higher values until a limit is arrived at.
Lists and labeled tree structures are supported as primitive data types.
No variable declaration is required rather these are deciphered at compile time otherwise an error is generated.
Variables with unchangeable values are initialized as follows
val<name> = <expresion>
fun<name> (<arguments>) = <body>
let <list of definitions> in <statements> end
val and fun statements are frequently used for definitions.
Function definitions can be nested, e.g, the body of a function p can contain a let-statement which includes the definition of another nested function q.
q can have function definitions within its body which leads to deep nesting of functions.
We define nesting depth 1 as the nesting depth for procedures that are not nested in any other procedures, e.g, all C functions have a nesting depth 1.
If a procedure p is defined within another procedure with a nesting depth i we have a nesting depth of i+1 for procedure p.
An example of quickSort algorithm in ML
1. fun sort(input_file, output_file) =
2. val a = array(11, 0);
3. fun read(input_file( = ...
4. ... a ...;
5. fun exchange(i, j) =
7. fun quiskSort(m, n) =
8. val v = ...;
9. fun partition(y, z) =
10. ... a ... v ... exchange ...
11. ... a ... v ... partition ... quiskSort
12. ... a ... read ... quickSort ...
At line (1) the sort function reads an array a of nine integers and sorts them.
It is the outermost function and has a nesting of depth 1.
At line (2), we have defined an array a of nine integers by declaring array as the first argument stating that we want an array to have 11 elements, the second argument states that all array elements will initially have a value of 0 which is an integer so we don't have to declare a type.
Functions at line (3) and (5) have a nesting depth of 2 since they are defined within a function with a nesting depth of 1.
These functions can access the array and can change the values of a, keep in mind that in ML, array accesses can violate the functional nature of the language.
From lines (7) through (11) we see the working of quickSort algorithm, v , the pivot element is declared at line (8).
The partition function at line(10) has a nesting depth of 3 since it's defined inside a function with a nesting depth of 2.
It accesses both the array a and the pivot element, it also calls the function exchange.
Line (11) we see quickSort function accessing variables a, v, partition function and itself recursively.
Line (12) states that the outer function sort accesses a and calls read and quickSort procedures.
We can directly implement the normal static-scope rule for nested functions by adding a pointer - access link, to each activation record.
That is, if a procedure p is nested within procedure q in the source code, then the access link in any activation will point to the most recent activation of q.
It is important for the nesting depth q to be one less than the nesting depth of p so that the access links form a chain from the activation record at the top of the stack to sequences of activations located at progressively lower depths.
Along this formed chain are all the activations whose data and procedures are accessible to the currently executing procedure.
Now, supposing a procedure p at the top of the stack is at a nesting depth
{n}_{p}
and p needs to access x - an element defined within a procedure q surrounding p and has the nesting depth
{n}_{p}
, We start the activation record for p at the top of the stack and follow the access link
{n}_{p}
{n}_{q}
times from the activation record for q.
This will always be the highest(most recent) activation record for q that currently appears on the stack and it will contain x.
The compiler will know the layout of activation records and thus x will be found at some fixed offset from the position of q's activation record.
Access links for finding non-local data for the quickSort algorithm
We have represented function names using their first letters and shown some of the data that might appear in various ARs as well as access links for each activation.
The image below shows the state after sort function calls read() which loads input into the array and quickSort(1, 9) to sort the array.
Access link from quickSort(1,9) will point to the activation record for sort since sort is more closely nested to the function surrounding quickSort().
The recursive calls to quickSort(1, 3) are followed by a call to partition which in turn calls exchange.
quickSort(1, 3)'s access link points to sort just like the access link for quickSort(1, 9).
The access link for exchange bypasses the activation records for quickSort and partition since exchange is nested immediately in sort.
This arrangement is allowed since exchange need access to the array a and the two elements(i, j) it must swap.
When a procedure q calls a procedure p explicitly, two cases exists;
The procedure p is at a higher nesting depth than q, Then p must be defined within q or the call by q would not be at a position that is within the scope of the procedure name p and therefore nesting depth of p is exactly one greater than that of q and access link from p must lead to q.
e.g from the quickSort() example, the call of quickSort() by sort() to set up the access links from 1 and the call to partition by quickSort() to create 2(right image).
The nesting depth
{n}_{p}
of p is less than or equal to the nesting depth of
{n}_{q}
of q.
In the order for the call within q to be in the scope of name p, procedure q must be nested within some procedure r while p is a procedure defined immediately within r.
The top activation record for r is therefore found by following the chain of access links starting from the activation record for q for
{n}_{q}
{n}_{p}
+ 1 hops.
The access link for p must go to this activation of r.
Recursive calls will be included where p = q, in this case the chain of access links is followed for one hop while the access links for p and q remain the same.
e.g Call for quickSort(1,) by quickSort(1, 9) sets up 2(Right-hand side)
This also includes the case for mutually recursive calls whereby two or more procedures are defined within a common parent.
An equivalent way for discovering access links is to follow access links for
{n}_{q}
{n}_{p}
hops and copy the access link found in the record.
When a procedure p is passed to another procedure q as a parameter and q calls its parameter(calls p in the activation of q), it is possible that q does not know the context in which p appears in the program and therefore impossible for q to know how to set the access link for p.
We solve this problem by the following, when procedures are used as parameters, the caller needs to pass the name of the procedure-parameter along with the proper access links for the parameter.
The caller will always know the link because if p is passed by procedure r as an actual parameter, p must then be named accessible to r and therefore r can determine the access link for p as if p were being directly called by r
We have the function a which has nested functions b and c
Function b calls a function-valued parameter
Function c defined a function d and hen call b with the parameter d
fun a(x) =
fun b(f) =
... f ... ;
fun c(y) =
fun d(z) = ...
... b(d) ...
... c(1) ...
Actual parameters carrying access links with them
access-link approach to non-local data have a drawback that is the nesting depth gets large, we follow long chains of links to reach the needed data.
To improve on this, we can use an auxilliary array d which we call a display and will contain a pointer for each nesting depth.
We expect that at all times d[i] is a pointer to the highest activation record on the stack for any procedure at nesting depth i.
sort at depth 1 calls quickSort(1, 9) at depth 2,
The activation record for quicksort will have a place to store the old value of d[2] indicated as saved d[2], in this case this pointer is null since there was no prior activation record at depth 2.
quickSort(1,9) calls quickSort(1,3) and since the ARs for both calls are at depth2, we store the pointer to quickSot(1, 9) previously in d[2] in the record for quickSort(1, 3), d[2] now points to quickSort(1, 3).
partition at depth 3 is called again, we use slot d[3] in the display and point it to the AR for partition.
The record for partition has a slot for a former value of d[3] but there is none in this case therefore the pointer remains null.
partition calls exchange at depth 2 therefore its AR stores the old pointer d[2] which goes to the activation record for quickSort(1, 3)
From (4) We see the display d with d[1] holding a pointer to the activation record for sort, highest and only activation record for a nesting depth of 1.
d[2] holds a pointer to the activation record for exchange the highest record for functions with nested depth of 2.
d[3] points to partition, the highest record at depth 3.
An advantage of this is that if procedure p is executing and needs to access element x belonging to a procedure q, we look only at d[i] where i is the nesting depth of q and follow the pointer d[i] to the activation record for q where x will be found at a known offset.
The compiler will know i and therefore can generate code to access x using d[i] and x's offset from the top of the AR for q and therefore the code doesn't need to follow a long chain of access links.
We save previous values of display entries in new AR so as to maintain the display correctly.
If procedure p at depth
{n}_{p}
is called and its activation record is not the first on the stack for a procedure at that depth, the activation record for p will be required to hold previous values of d[
{n}_{p}
], while d[
{n}_{p}
] itself is set to point to the activation of p.
When p returns, its AR is removed from the stack and we restore d[
{n}_{p}
] to have its value prior to the call.
Accessing non-local data is compilicated in programming languages where a procedure is declared inside another procedure.
We have use ML functional programming language to demonstrate how we go about this access.
Compilers, principles, techniques and tools, 2nd edition Alfred V. Aho, Monica S. Lam, Ravi Sethi, Jeffrey D. Ullman.
Basics of Compiler Design Torben Ægidius Mogensen
Trace-Based garbage collection
Trace-based collection is garbage collection whereby garbage is collected periodically unlike basic gc where garbage is collected as it is created. In this article we discuss four algorithms for garbage collections and their properties.
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The multivariate gaussian is a crucial building block in machine learning, robotics, and mathematical finance applications. Here is just a small sample of applications.
However, probability density formula of a multivariate gaussian can be pretty intimidating.
p(x) = \frac 1 {2 \pi}^{k/2} \det(\Sigma)^{-1/2} e^{-\frac 1 2 (x - \mu)^T \Sigma^{-1} (x - \mu)}
While this expression may work for a mathematician, it can be a little overwhelming for other audiences. In this post I’ll provide an alternate rigorous, programmatic, definition of the multivariate gaussian in Python. We’ll then be able to use our mental Python interpreter to get intuition about the behavior of any multivariate gaussian.
I will assume familiarity with the following concepts.
the single dimensional gaussian, aka the normal random variable
expectation and covariance of random variables
vector and matrix operations, such as multiplication, and transpose
basic Python and numpy notation
We’ll also need the concept of simulation. I’ll say that a function f simulates a random variable F if there is no way to determine, given a list of outputs, whether the list was generated by repeated function calls to f or whether it came from independent realizations of F.
Now we’re ready to give a computational definition of an
dimensional multivariate gaussian.
Examine this simple program.
n = ... # Fill this in with a positive integer dimension, say n = 3
Return an n-dimensional column vector of independent standard random normals
return np.random.normal(size=(n,1))
def x(mu, M):
Return a random n-dimensional vector according to our special formula
mu -- an n-dimensional column vector
M -- an nxn matrix
return mu + np.matmul(M, z())
X
is an
-dimensional multivariate gaussian if and only if its distribution can be simulated by repeatedly calling the above function x(mu, M) with some constant inputs mu and M.
Now let’s break down what x(mu, M) does.
Fill an n-dimensional vector with independent standard random normals by calling z().
Then multiply the result by matrix M.
Then shift the result by mu.
This computational viewpoint is especially useful if M has a geometrical interpretation. For example, suppose we are working in two dimensions
n = 2
. If we graph the two columns of M, we can get a picture of how it warps the plane.
Successive calls to z() will generate a cloud of samples around the origin, with the cloud getting denser as we get closer to the origin. If we draw a circle around this cloud and multiply all points in the plane by M, we can see that we get a cloud in an elliptical shape.
All that remains is to shift this cloud of points by mu and we have a visualization for the two dimensional multivariate gaussian simulated by x(mu, M).
This ellipsoid representation, by the way, is exactly what we use in the self driving car industry to represent the uncertainty estimates for the positions of other cars.
Suppose we know that
X
is simulated by x(mu, M) (and therefore it is, by our definition, multivariate gaussian) but we want to convert to the popular mean
\mu
\Sigma
parameterization. It’s a simple matter of computing statistics.
First let’s compute the mean.
\mu = E[X]
= E[
x(mu, M)
]
= E[
mu + M z()
] =
mu + M
E[
z()
]
(factoring out constants)
=
0
E[
] = 0
=
Unsurprisingly, the mean
\mu
is mu. We could have guessed that by looking at the illustrations above.
Now let’s compute the covariance. But first a clarification about notation. Consider that np.matmul(z(), z().T) calls z twice and therefore generates two independent versions of z. How do we declare that we want the same sample of z both times? We can use an immediately invoked lambda expression (lambda Z: np.matmul(Z Z.T))(z()). (You might have seen this elsewhere called the IIFE pattern.) Okay we’re ready to compute the covariance.
\Sigma = E[(X - \mu)(X - \mu)^T]
= E[
(lambda X: (X - mu)(X - mu).T)(x(mu, M))
]
= E[
(lambda Z: M Z (M Z).T)(z())
]
= E[
(lambda Z: M Z Z.T M.T)(z())
]
=
E[
(lambda Z: Z Z.T)(z())
]
M.T
=
I
M.T (using
E[
] = I
— diagonal terms are variances and cross terms are covariances of standard independent normals)
=
M M.T
So the covariance
\Sigma
is M M.T and we get this result.
Result 1. If
X
is a gaussian simulated by x(mu, M), it has mean mu and covariance M M.T.
It’s actually more likely you’ll need to go backwards. For example, if you encounter a multivariate gaussian in the wild with the
\mu
\Sigma
parameterization and you need want to visualize it as an ellipsoid using the mu, M parameterization. For that we usually use the eigendecomposition
\Sigma = Q \Lambda Q^T
This decomposition is extremely important in linear algebra, and I would encourage you to read up on it later. If we start with any covariance matrix
\Sigma
, the eigendecomposition has the following properties.
\Lambda
is a diagonal matrix with positive entries
Q
is a high dimensional rotation matrix (all columns are mutually orthogonal)
\Lambda^{\frac 1 2}
as the diagonal matrix whose entries are the square root of
\Lambda
’s entries, we can see from Result 1 that calling x(mu=
\mu
, M=
Q \Lambda^{\frac 1 2}
) will simulate a multivariate gaussian with mean
\mu
and covariance M M.T
= Q \Lambda^{\frac 1 2} (Q \Lambda^{\frac 1 2})^T = Q \Lambda Q^T = \Sigma
. So we have the following result.
Result 2. A multivariate gaussian
X
\mu
\Sigma
is simulated by x(mu=
\mu
Q \Lambda^{\frac 1 2}
Q \Lambda Q^T
is the eigendecomposition of
\Sigma
Let’s break down the action of
Q \Lambda^{\frac 1 2}
into two steps, the action of
\Lambda^{\frac 1 2}
and then the action of
Q
\Lambda^{\frac 1 2}
is a diagonal matrix so it acts in an extremely simple way — by stretching the
i^{th}
axis by the scaling factor given in its
i^{th}
diagonal entry.
\Lambda^{\frac 1 2} x = \begin{bmatrix} \sigma_1 && 0 && 0 \\ 0 && \sigma_2 && 0 \\ 0 && 0 && \sigma_3 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} \sigma_1 x_1 \\ \sigma_2 x_2 \\ \sigma_3 x_3 \end{bmatrix}
Q
is a rotation matrix whose action is also easily interpretable from inspecting its columns using column oriented multiplication.
So using our mental Python interpreter as before, we can visualize the samples x(mu=
\mu
Q \Lambda^{\frac 1 2}
) by imagining a cloud of samples contained in a sphere, then warping the sphere into an ellipsoid whose axes lengths are given by
\Lambda^{\frac 1 2}
, and then shifting this whole ellipsoid by
\mu
If you are fortunate enough to be working at a robotics company, you can do a code search for your gaussian visualization routines and I bet you will find something like the following.
def vis_gaussian(mean, covar):
"""Visualize a 3d gaussian"""
eigen_vals, eigen_vects = np.linalg.eig(covar)
plot_ellipsoid(
center=mean,
major_axes=eigen_vects,
axes_lengths=np.sqrt(eigen_vals))
Now I hope whenever you see
X \sim N(\mu, \Sigma)
you’ll find it helpful to mentally replace it with
X =
x(mu=
\mu
Q \Lambda^{\frac 1 2}
) and simulate it with your mental Python interpreter.
← We Learned the Wrong Way to Matrix Multiply
Vtubing and Streaming Quick Start →
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Revision as of 04:18, 26 February 2013 by Gmaxwell (talk | contribs) (→GTK-Bounce: fix another trac link)
{\displaystyle \ squarewave(t)={\begin{cases}1,&|t|<T_{1}\\0,&T_{1}<|t|\leq {1 \over 2}T\end{cases}}}
{\displaystyle {\begin{aligned}\ squarewave(t)={\frac {4}{\pi }}\sin(\omega t)+{\frac {4}{3\pi }}\sin(3\omega t)+{\frac {4}{5\pi }}\sin(5\omega t)+\\{\frac {4}{7\pi }}\sin(7\omega t)+{\frac {4}{9\pi }}\sin(9\omega t)+{\frac {4}{11\pi }}\sin(11\omega t)+\\{\frac {4}{13\pi }}\sin(13\omega t)+{\frac {4}{15\pi }}\sin(15\omega t)+{\frac {4}{17\pi }}\sin(17\omega t)+\\{\frac {4}{19\pi }}\sin(19\omega t)+{\frac {4}{21\pi }}\sin(21\omega t)+{\frac {4}{23\pi }}\sin(23\omega t)+\\{\frac {4}{25\pi }}\sin(25\omega t)+{\frac {4}{27\pi }}\sin(27\omega t)+{\frac {4}{29\pi }}\sin(29\omega t)+\\{\frac {4}{31\pi }}\sin(31\omega t)+{\frac {4}{33\pi }}\sin(33\omega t)+\cdots \end{aligned}}}
https://trac.xiph.org/browser/trac/Xiph-episode-II/bounce/
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Enthalpies of Reaction - Course Hero
General Chemistry/Enthalpy and Bond Strength/Enthalpies of Reaction
\Delta H=H_{\rm{products}}-H_{\rm{reactants}}
The enthalpy change that occurs during a reaction is called the enthalpy, or heat, of reaction. Reaction enthalpies of common reactions have been experimentally measured under constant pressure. Consider the reaction between hydrogen and oxygen to make water:
2{\rm H}_2(g)+{\rm O}_2(g)\rightarrow2{\rm H}_2{\rm O}(g)
When 2 moles of hydrogen react with 1 mole of oxygen to produce 2 moles of water, this reaction releases 483.6 kJ of heat. The change in enthalpy is equal to the enthalpy of the products minus the enthalpy of the reactants:
\Delta H=H_{\rm{products}}-H_{\rm{reactants}}
This reaction releases heat. This means the products have less energy than the reactants, and
\Delta H
will be a negative value.
\Delta H
is –483.6 kJ for this reaction. This sign convention holds true for all reactions: If a reaction releases heat (exothermic), then it has a negative
\Delta H
. If a reaction absorbs heat (endothermic), then
\Delta H
is positive. Reaction enthalpies are proportional to the masses of the substances involved. Enthalpy is an extensive property, a property that is determined by the amount of matter in a system. If 4 moles of hydrogen react with 2 moles of oxygen to produce 4 moles of water, the energy released would be
483.6\times2=967.2\;{\rm{kJ}}
If the chemical equation is reversed, the sign of the enthalpy of reaction is reversed too. Water can be broken down into its component elements through electrolysis.
2{\rm H}_2{\rm O}(g)\rightarrow2{\rm H}_2(g)+{\rm O}_2(g)\;\;\;\;\;\Delta H=483.6\;{\rm{kJ}}
In this reaction, 483.6 kJ of energy must be provided to break down 2 moles of water into 2 moles of hydrogen and 1 mole of oxygen. Note that the
\Delta H
is positive, indicating an endothermic reaction. In practical applications of this reaction, energy is provided as electricity, not heat. Enthalpy change also depends on the phase of the reactants and the products. Changing phases requires or releases energy. Changing 1 mole of liquid water into 1 mole of water vapor, for example, absorbs 44 kJ of energy.
{\rm H}_2{\rm O}(l)\rightarrow{\rm H}_2{\rm O}(g)\;\;\;\;\;\Delta H=44\;{\rm{kJ}}
Calculation of an Enthalpy Change for the Combustion of Methane
Determine the amount of energy released during the combustion of 12.0 g methane (CH4). The reaction energy is
\Delta H=-802.3\;{\rm{kJ}}
{\rm{CH}}_4(g)+2{\rm{O}}_2(g)\rightarrow{\rm{CO}}_2(g)+2{\rm{H}}_2{\rm{O}}(g)\;\;\;\;\;\;\Delta{H}=-802.3\;{\rm{kJ}}
The equation provides enthalpy data for combustion of 1 mole of methane. To calculate how much energy is released by 12.0 grams of methane, convert 12.0 grams of methane gas to moles. First calculate the molar mass of methane. Use the periodic table to determine the atomic mass of each atom. Then, multiply the corresponding molar mass of each atom by the number of atoms shown in the formula for methane.
\begin{aligned}\text{molar mass of }\rm{CH_4}&=(1)(12.01\,{\rm{g/mol}})+(4)(1.01\;{\rm{g/mol}})\\&=16.05\;{\rm{g/mol}}\end{aligned}
Next, use the molar mass as a conversion factor when determining the energy released by 12.0 g of methane.
\begin{aligned}\Delta H&=(\Delta H_{\rm{mole}})(\text{moles }\rm{CH_4})\\&=\!\left(\displaystyle\frac{-802.3\;{\rm{kJ}}}{1\;{\rm{mol\;CH}}_{4}}\right)\!(12.0\;{\rm{g\;CH}}_{4})\!\left(\displaystyle\frac{1\;{\rm{mol\;CH}}_{4}}{16.0\;{\rm{g\;CH}}_{4}}\right)\\&=-602\;{\rm{kJ}}\end{aligned}
Determine Enthalpy Change When Forming Ethyne
Ethyne (C2H2), also known as acetylene, forms by adding hydrogen gas (H2) to solid carbon. This reaction is carried out in a constant-pressure calorimeter. When 18.0 grams of carbon reacts with excess hydrogen, 170.0 kJ of heat is absorbed. What is
\Delta H
for this reaction?
2{\rm C}(s)+{\rm H}_{2}(g)\rightarrow{\rm C}_{2}{\rm H}_{2}(g)
The chemical equation shows what happens when two moles of carbon react with one mole of hydrogen. To solve the problem, first calculate the number of moles of carbon in 18.0 g of carbon.
\begin{aligned}\text{moles of carbon}&=\frac{\text{mass of carbon}}{\text{molar mass of carbon}}\\&=\frac{18.0\;{\rm{g}}}{12.01\;{\rm{ g/mol}}}\\&=1.499\;{\rm{mol\;C}}\end{aligned}
Convert the measured heat per 1.499 mol C to
\Delta H
\begin{aligned}\Delta{H}&=\left(\frac{170.0\;{\rm{kJ}}}{1.499\;{\rm{mol\;C}}}\right)\!\left(\frac{2\;{\rm{mol\;C}}}{1\;{\rm{mol\;C}}_{2}{\rm{H}}_{2}}\right)\\&=227\;{\rm{kJ}}\end{aligned}
The fraction is multiplied by the 2 mol of carbon that is required to produce each mole of C2H2. The final answer is expressed in kJ rather than kJ/mol because
\Delta H
is assumed to be per mole.
Scientists measure the enthalpies of reactions experimentally. Consider the combustion of methane:
{\rm{CH}_{4}}(g)+2{\rm{O}_{2}}\;(g)\rightarrow{\rm{CO}_{2}}(g)+2{\rm{H}_{2}}{\rm{O}}(g)
At 1 atm pressure and 25°C, this reaction has a
\Delta H
of −802.3 kJ, as given in the question. If the same reaction is carried out at 1,000°C and 1 atm pressure, the
\Delta H
changes to −792.4 kJ. Similar differences occur when enthalpy measurements are taken under different conditions. A standardization between enthalpy measurements is needed. The standard state is the set of specific conditions under which reactions are measured, typically 0°C and 1 atm. Recall that 273.15 K is equivalent to 0°C. Standard state is explicitly defined for different substances as follows:
For solids and liquids, standard state is a stable state under 1 atm pressure and an often-specified temperature of 25°C.
For gases, standard state is a gas with a pressure of 1 atm pressure and an often-specified temperature of 25°C.
For solutions, standard state is 1.0 M concentration under 1 atm pressure at an often-specified temperature of 25°C.
When all the reactants and products are at their standard state for a specific reaction, the enthalpy for the reaction is called the standard enthalpy of reaction (
\Delta H^\circ
<Enthalpy>Hess’s Law
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The life expectancy of a male whose current age is
The life expectancy of a male whose current age is x years old is f(x)=0.0069502x^2−1.6357x+93.76 (60<=x<=75) years
Advances in medical science and healthier lifestyles have resulted in longer life expectancies. The life expectancy of a male whose current age is x years old is
f\left(x\right)=0.0069502{x}^{2}-1.6357x+93.76\text{ }\left(60\le x\le 75\right)
years. What is the life expectancy of a male whose current age is 65? A male whose current age is 75?
16.8 years for male who is 65 years old and 10.18 years for a male aged 75 years
\begin{array}{|ccc|}\hline & Advanced& Beginning\\ Brass& 12& 28\\ Strings& 22& 25\\ Woodwinds& 18& 30\\ \hline\end{array}
\underset{n\to \mathrm{\infty }}{lim}\frac{{a}_{2n}}{{a}_{n}}<1/2
\sum {a}_{n}
\underset{n\to \mathrm{\infty }}{lim}\frac{{a}_{2n+1}}{{a}_{n}}>1/2
\sum {a}_{n}
{a}_{n}=\frac{{n}^{\mathrm{ln}n}}{{\left(\mathrm{ln}n\right)}^{n}}
\frac{{a}_{2n}}{{a}_{n}}\to 0
n\to \mathrm{\infty }
Let Z be a cycle of generalized eigenvectors of a linear operator T on V that corresponds to the eingenvalue lambda.Prove that span(Z) is a T-invariant subspace of V.
The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence.
The test states that if
\underset{n\to \mathrm{\infty }}{lim}\frac{{a}_{2n}}{{a}_{n}}<1/2
\sum {a}_{n}
\underset{n\to \mathrm{\infty }}{lim}\frac{{a}_{2n+1}}{{a}_{n}}>1/2
\sum {a}_{n}
{a}_{n}=\frac{1}{1+x}\frac{2}{2+x}\dots \frac{n}{n+x}\frac{1}{n}=\frac{\left(n-1\right)!}{\left(1+x\right)\left(2+x\right)\dots \left(n+x\right)}
{a}_{2n}/{a}_{n}\le {e}^{-x/2}/2
For which x > 0 does the generalized ratio test imply convergence of
\sum _{n=1}^{\mathrm{\infty }}{a}_{n}
A city recreation department offers Saturday gymnastics classes for beginning and advanced students. Each beginner class enrolls 15 students, and each advanced class enrolls 10 students. Available teachers, space, and time lead to the following constraints. ∙∙ There can be at most 9 beginner classes and at most 6 advanced classes. ∙∙ The total number of classes can be at most 7. ∙∙ The number of beginner classes should be at most twice the number of advanced classes.
a. What are the variables in this situation?
b. Write algebraic inequalities giving the constraints on the variables.
c. The director wants as many children as possible to participate. Write the objective function for this situation.
d. Graph the constraints and outline the feasible region for the situation.
e. Find the combination of beginner and advanced classes that will give the most children a chance to participate.
f. Suppose the recreation department director sets new constraints for the schedule of gymnastics classes. ∙∙ The same limits exist for teachers, so there can be at most 9 beginner and 6 advanced classes. ∙∙ The program should serve at least 150 students with 15 in each beginner class and 10 in each advanced class. The new goal is to minimize the cost of the program. Each beginner class costs $500 to operate, and each advanced class costs $300. What combination of beginner and advanced classes should be offered to achieve the objective? .
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Delay line - MATLAB - MathWorks France
'TimeDelay'
'Z0'
Represent Delay Lines
Use the delay class to represent delay lines that are characterized by line loss and time delay.
h = rfckt.delay
h = rfckt.delay(Name,Value)
h = rfckt.delay returns a delay line object whose properties are set to their default values.
h = rfckt.delay(Name,Value) sets properties using one or more name-value pairs. For example, rfckt.delay('Loss',2)creates an RF delay line with a line loss of 2 dB. You can specify multiple name-value pairs. Enclose each property name in a quote. Properties not specified retain their default values.
Computed S-parameters, noise figure, OIP3, and group delay values, specified as a rfdata.data object. For more information refer, Algorithms.
'Loss' — Line loss value
0 | positive scalar in dB
Line loss value, specified as a positive scalar in dB. Line loss is the reduction in strength of the signal as it travels over the delay line .
'TimeDelay' — Amount of time delay
1.0000e-012 | scalar in seconds
Amount of time delay introduced in the line, specified as a scalar in seconds.
'Z0' — Characteristic impedance
50 | scalar in ohms
Characteristic impedance of the delay line, specified as a scalar in ohms.
Represent delay lines that are characterized by line loss and time delay using rfckt.delay.
del=rfckt.delay('TimeDelay',1e-11)
del =
rfckt.delay with properties:
TimeDelay: 1.0000e-11
Name: 'Delay Line'
The analyze method treats the delay line, which can be lossy or lossless, as a 2-port linear network. It computes the AnalyzedResult property of the delay line using the data stored in the rfckt.delay object properties by calculating the S-parameters for the specified frequencies. This calculation is based on the values of the delay line's loss, α, and time delay, D.
\left\{\begin{array}{c}{S}_{11}=0\\ {S}_{12}={e}^{-p}\\ {S}_{21}={e}^{-p}\\ {S}_{22}=0\end{array}
Above, p = αa + iβ, where αa is the attenuation coefficient and β is the wave number. The attenuation coefficient αa is related to the loss, α, by
{\alpha }_{a}=-\mathrm{ln}\left({10}^{\alpha /20}\right)
\beta =2\pi fD
where f is the frequency range specified in the analyze input argument freq.
rfckt.amplifier | rfckt.cascade | rfckt.coaxial | rfckt.cpw | rfckt.datafile | rfckt.hybrid | rfckt.hybridg | rfckt.mixer | rfckt.microstrip | rfckt.passive | rfckt.parallel | rfckt.parallelplate | rfckt.rlcgline | rfckt.series | rfckt.seriesrlc | rfckt.shuntrlc | rfckt.twowire | rfckt.txline
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All optimizations in a compiler depend on data-flow analysis.
We have three instances of data-flow problems, reaching definitions, live variables and available expressions.
3.1. Statement semantics.
3.2. Flow of control.
Schemas on basic blocks.
Reaching definitions.
Live-variable analysis.
Data-flow analysis is the body of techniques involved in deriving information about the flow of data along program execution paths.
For example, to implement global common sub-expression elimination, we are required to determine whether two textually identical expressions evaluate the same value along any execution path of the program.
The execution of a program is viewed as a series of transformations of the program state that consists of values of all variables in the program including the ones associated with stack frames below the top of the run-time stack.
When an intermediate-code statement is executed, the input state is transformed into a new output state.
To analyze this phenomenon, we consider all possible sequences of paths the program takes during execution after which we extract the information we need to solve a data-flow analysis problem.
The flow graph tells us the following about an execution path:
Within a basic block, the program point after a statement is the same as the program point before the next statement.
If there is an edge from a block
{\mathrm{B}}_{1}
{\mathrm{B}}_{2}
, then the program point after the last statement of B1 is followed immediately by the program point before the first statement of B2.
We therefore define an execution path from point
{\mathrm{p}}_{1}
{\mathrm{p}}_{n}
as the sequence of points p1, p2, ..., pn such that for each i = 1, 2, ..., n-1, either;
{\mathrm{p}}_{i}
is the point that immediately precedes a statement and
{\mathrm{p}}_{i}
+ 1 the point immediately following the same statement.
pi is the end of a block and pi + 1 is the beginning of the following block.
Generally, there exists an infinite number of possible execution paths through a program and there is no finite upper bound on the length of an execution path.
Analysis of a program summarizes all possible states of the program that can occur into a finite set of facts.
Note that, different analyses may choose to abstract different information and therefore there is no perfect representation of a state in data-flow analysis.
For every data-flow analysis, a program is assigned a data flow value representing an abstraction of the set of all possible program states that can be observed for that point.
This set of values is the domain for this application. E.g, the domain of data-flow values for reaching definitions is the set of all subsets of definitions in the program.
Now, we want to associate with each point in the program the set of definitions that can reach that point.
As the choice of abstraction depends on the analysis goal, we only track relevant information.
We denote data-flow values before and after each s statement by IN[s] and OUT[s] respectively.
Now, the data-flow problem is to find a solution to a set of constraints on the IN[s] and OUT[s] for all statements s.
We have two sets of constraints, the first is based on statements semantics of the statements while the second is based on the flow of control.
Statement semantics.
An example: Assuming the analysis involves determining the constant value of variables at points. If a variable a has a value v before executing statement b = a, both a and b will have value v after the statement.
This relationship is referred to as a transfer function.
Such a function can be of two flavors:
For information propagated forward along execution paths, the transfer function of a statement s is denoted by
{\mathrm{f}}_{s}
, it takes the data-flow value before the statement an produces a new data-flow value after the statement:
OUT[s] =
{\mathrm{f}}_{s}
(IN[s])
For information flowing backwards up the execution paths, the transfer function
{\mathrm{f}}_{s}
for a statement s converts a data-flow value after the statement to a new data-flow value before the statement:
IN[s] =
{\mathrm{f}}_{s}
(OUT[s])
Within a basic block, the control flow is simple, if a block B has statements s1, s2, ..., sn, then the control-flow value out of
{\mathrm{s}}_{i}
is similar to the control-flow value into
{\mathrm{s}}_{i+1}
IN[
{\mathrm{s}}_{i+1}
] = OUT[
{\mathrm{s}}_{i}
], for all i = 1, 2, ..., n-1.
Control flow edges between basic blocks create complex constraints between the last statement of a basic block and the first statement of the following block. E.g is we want to collect definitions reaching a point in a program, the set of definitions after the last statement of the block is the union of definitions after the last statements of each predecessor block.
A data-flow schema involves data-flow values at each point in the program.
Control flows from the start to the end of a block without interrupting a branch.
We restate this schema in terms of these data-flow values that enter and leave the blocks.
We also represent these data-flow values immediately before and after basic block B by IN[B] and OUT[B] respectively.
Constraints involving IN[B] and OUT[B] are derived from those involving IN[s] and OUT[s] for various statements s in B as follows.
Assuming block B has statements s1, ..., sn, if
{\mathrm{s}}_{1}
is the first statement, of block B, then IN[B] = IN[
{\mathrm{s}}_{1}
In a similar way, of
{\mathrm{s}}_{n}
is the last statement of block B denoted by
{\mathrm{f}}_{B}
, is derived by composing the transfer functions of the statements in the block.
{\mathrm{f}}_{si}
is the transfer function of statement si, then:
{\mathrm{f}}_{B}
{\mathrm{f}}_{sn}
{\mathrm{f}}_{s2}
{\mathrm{f}}_{s1}
The relationship between the start and end of the block is
OUT[B] =
{\mathrm{f}}_{B}
(IN[B])
We can rewrite the constraints between blocks dur to control flow by substituting IN[B] and OUT[B] for IN[
{\mathrm{s}}_{1}
] and OUT[
{\mathrm{s}}_{n}
] respectively.
Unlike linear arithmetic equations, data-flow equations don't have a unique solution and therefore the goal is to find the most precise solution that satisfies two sets of constraints.
We need a solution in addition to supporting code improvements, it doesn't try to justify unsafe transformations, this excludes those transformations that change what a program computes.
This is a common and useful data-flow schema.
The idea is this, by knowing the point in the program where a variable x may be defined when control reaches point p, we can gather a lot of information about x.
In addition to a compiler being able to determine if x is a constant at point p in instances, a debugger, can tell whether it is possible for constant x to be undefined if it is used at point p.
A 'definition of variable x' is the statement assigning or may assign a value to x.
We state that a definition d reaches a point p if there exists a path from the point following d to p such that definition d is not killed along the path, on the other hand, If there exists another definition of x that is along this path we kill variable x.
And so, If the definition d of variable x reaches point p, then d might be at the place at which the value of x that was used at p was last defined.
Procedure parameters, array accesses, and indirect references all have one thing in common, they have aliases therefore it is difficult to tell if a statement refers to a variable x.
We also get to kill a definition of variable x if there exists another definition of x along the path.
We kill a definition variable, that is if there exists another path definition of x along this path.
Program analysis is conservative if we don't know whether statement s assigns a value to x, we assume that it may assign to it, that is, variable x after statement s might either have its original value before s or a new value created by s.
Code-improving transformations depend on the information computed in the direction opposite to the current flow of control.
In live variable analysis, we ask, for a variable x and a point p, can the value of x at p be used along the same path in the flow graph starting at p?
If the value x can be used, we refer to x as live otherwise x is dead.
Live-variable analysis is used in register allocation for basic blocks, where after
a value is computed in a register and presumably used within a block, there is no need to store that value in the register if it is dead at the end of the block.
Also if all available registers are full, and we need another register, we will preferably use a register containing a dead value since the said value does not have to be stored.
Given an expression, x + y is available at point p if, for every path from the entry node to p, the latter evaluates to x + y, and after the last similar evaluation before reaching p, there aren't any subsequent assignments to x or y, therefore for every available-expressions data-flow schema, we state that a block kills expression x + y if it assigns x or y and doesn't subsequently need to recompute x + y.
A block generates expression x + y if it definitely evaluates x + y and doesn't subsequently define x or y.
The idea of killing and generating an available expression is not similar to that of reaching definitions. Nevertheless, they behave essentially as they do for reaching definitions.
To conclude, available expressions can also be used in detecting global common subexpressions.
All optimizations in the compiler depend on data-flow analysis.
We have discussed three instances of data-flow problems namely; reaching definitions, live variables and available expressions.
The definition of each of the above problems is given by the domain of data-flow values, the direction the data is flowing, the family of transfer functions, the boundary condition, and the meet operator.
Modern Compiler Design. Chapter 5, Dick Grune, Koen Langendoen.
Compilers Principles, Techniques, & Tools. Chapter 9, Ravi Sethi, Jeffrey D. Ullman.
Systemctl and Systemd in Linux
The systemctl command is a systemd utility used to manage services, get information about service unit files and service states and therefore a useful utility to know for managing services on the server while systemd is an array of components for Linux OS.
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Conic Sections, Popular Questions: CBSE Class 11-commerce SCIENCE, Science - Meritnation
Find the equation of tangent plane at (alpha, beta, gamma) to the concoid ax2 + by2 + cz2 = 1.
State whether the following is True or False: The line x +y = 0 intersects the circle 2 x ^2+ y ^2=1 in two points.
Find the equation of the tangent plane at a point( x1;y1;z1 )of the conicoid given in figure
Nida Firdous asked a question
Show that tangents from a focus to a conic satisfy the conditions for a circle
Himalayan Progressive School asked a question
Find the number of distinct real tangents that can be drawn from (0,-2) to the para bola y^2=4x. Also find the slope of tangents.
I have already tried this sum, and I got the answer. But I was able to find only one slope and the other one is infinity but how?
Manali Shetye asked a question
Bharat Rajani asked a question
Ishank Aggarwal asked a question
The shortest distance between the circles x^2 + y^2 =144 and (x-3)^2 + (y-4)^2 =25 is-
If the axes are rectangular, then find the locus of the equal conjugate diameters of ellipsoid, x2/a2 + y2/b2 + z2/c2 = 1
Ranjeeta Yadav asked a question
Ritwik Ranjan asked a question
Three normals to y2 = 4x pass through the point (15,12). Show that one of the normals is given by y = x - 3 find the equations of the others.
Chand Sood asked a question
Find the centre and the radius of the equation 3x2+ 3y2+ 6x -4y -1 =0 of the circle.
Question) Find the eqn of the Director - circle of the conic
\frac{l}{r}=1+e \mathrm{cos} \theta &
find the eqn to the locus of the foot of the perpendicular form the focus of the above conic on the tangent.
Abhigyan Biswas asked a question
Polar of any point P w.r.t two given circles meet at Q. Prove that radical axis of circles bisect the line segment PQ
Find the equation of the hyperbola whose foci are (0,12) and (0,-12) and length of the latus rectum is 36.
Olivia Paul asked a question
Shashwat Rakesh Dutt asked a question
The circle passing through (1,-2) and touching the axis of x at (3,0) also passes through the point:
Please help me with C-6 sum
Arkajyoti Purkayastha asked a question
if the chord of contact of tangents from a point P(h,k) to the circle x2 + y2 = a2 touches the circle x2 + (y-a)2 = a2, then locus of P is
Harshil Marwah asked a question
A normal to the hyperbola, 4x2 - 9y2 = 36 meets the co-ordinate axes x and y at A and B, respectively. If the parallelogram OABP (O being the origin) is formed, then the locus of P is:
1) 9x2 - 4y2 = 169
3) 9x2 + 4y2 = 169
4) 4x2 + 9y2 = 121
Siva Raj asked a question
find the equation of hyperbola where foci are (0,+-12) and the length of the latus rectum is 36
Kshitiz Anand asked a question
Q no9 ans is option b
A tangent to the ellipse 4x2+9y2=36 is cut by the tangent at the extremities of the major axis at T and T'. The circle on TT' as diameter passes through the point
(A) (0-
\sqrt{}
5) (B) (
\sqrt{}
5, 0 )
Find the distance between the chords of contact of the tangent to the circle x2 +y2 +2gx+2fy+c=0 from the origin and the point (g,f) .
A radio telescope has a parabolic dish with diameter of 100 metres. The collected radio signals are reflected to one collection point, called the "focal" point, being the focus of the parabola, If the focal length is 45 metres, find the depth of the dish, rounded to one decimal place.
{x}^{2}+{y}^{2}-8y-8=0
5x-2y=2
bilzkashif... asked a question
{x}^{2}={y}^{2} \mathrm{and} {\left(x-A\right)}^{2}+{y}^{2}=1
Jesal asked a question
Nishtha Gahlot asked a question
The locus of mid points of the portions of the tangents to the ellipse x2/a2 +y2/b2=1 included between the axes is the curve?
Prem Shankar asked a question
if a circle passes through the points of intersection of lines 2x - y +1 = 0 and x + Ly - 3 = 0 with the axes of reference then value of L is
Q no. 17.
Q.17. The radius of the circle whose two normals are represented by the equation
{x}^{2}-5xy-5x+25y=0
and which touches externally the circle
{x}^{2}+{y}^{2}-2x+4y-4=0
will be-
Find the equation of the ellipse with eccentricity 3/4 , foci on y- axis , center at
the origin and passes through the point ( 6, 4)
Anand Dubey asked a question
find the latus rectum,eccentricity and coordinates of the foci of the ellipse x2+3y2=k2.
Find the equation of the parabola whose focus is at (-6,6) and vertex is at (-2,2).
Sr asked a question
A triangle has two of its sides along the axes, its third side touches the circle x^2 +y^2 -4x -4y +4= 0. Prove that locus of circumcentre of triangle is 2(x+ y-1) =xy
Madhurima Purkait asked a question
Varad asked a question
Raghav Singh asked a question
Please explain the last 7 statements .What is delta trying to represent here?
equation of circle touching |x| + |y| = 4
Shagun Mishra asked a question
Tangents are drawn to the ellipse x^2(square) +(plus)2y^2(square)=(equal to) 4 from any arbitrary point on the line X +(plus) Y equal to 6 the corresponding chord of contact will always passes through the fixed point
I HAVE UPLOADED THR SOLUTION I HAVE A DOUBT IN THE SOLUTION. WHAT TO DO AFTER THE UNDERLINED PART.
Ritabh Ranjan asked a question
Prerna Jain asked a question
Vanshika Sachdeva asked a question
Please solve Q3 and please solve it full
Q3. Find the equation of the parabola that satisfies the (i) Focus (6, 0) ; directrix x = – 6 (ii) Focus (0,–3); directrix y = 3 (iii) vertex (0,0) ; focus (3, 0) (iv) vertex (0, 0) ; focus (–2, 0) (v) vertex (0,0), Passing through (2, 3) and axis is along x-axis (vi) vertex (0, 0), passing through (5, 2) and symmetric with respect to y-axis.
Find the equation of circle which touches both the axis and passes through the point (2,1).
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Adaptive estimation of a quadratic functional of a density by model selection
We consider the problem of estimating the integral of the square of a density
from the observation of a
n
sample. Our method to estimate
{\int }_{ℝ}{f}^{2}\left(x\right)\mathrm{d}x
is based on model selection via some penalized criterion. We prove that our estimator achieves the adaptive rates established by Efroimovich and Low on classes of smooth functions. A key point of the proof is an exponential inequality for
U
-statistics of order 2 due to Houdré and Reynaud.
Mots clés : adaptive estimation, quadratic functionals, model selection, Besov bodies, efficient estimation
author = {Laurent, B\'eatrice},
title = {Adaptive estimation of a quadratic functional of a density by model selection},
AU - Laurent, Béatrice
TI - Adaptive estimation of a quadratic functional of a density by model selection
Laurent, Béatrice. Adaptive estimation of a quadratic functional of a density by model selection. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 1-18. doi : 10.1051/ps:2005001. http://www.numdam.org/articles/10.1051/ps:2005001/
[1] P. Bickel and Y. Ritov, Estimating integrated squared density derivatives: sharp best order of convergence estimates. Sankhya Ser. A. 50 (1989) 381-393. | Zbl 0676.62037
[2] L. Birgé and P. Massart, Estimation of integral functionals of a density. Ann. Statist. 23 (1995) 11-29. | Zbl 0848.62022
[3] L. Birgé and P. Massart, Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli 4 (1998) 329-375. | Zbl 0954.62033
[4] L. Birgé and Y. Rozenholc, How many bins should be put in a regular histogram. Technical Report Université Paris 6 et 7 (2002).
[5] J. Bretagnolle, A new large deviation inequality for
U
-statistics of order 2. ESAIM: PS 3 (1999) 151-162. | Numdam | Zbl 0957.60031
[6] D. Donoho and M. Nussbaum, Minimax quadratic estimation of a quadratic functional. J. Complexity 6 (1990) 290-323. | Zbl 0724.62039
[7] S. Efroïmovich and M. Low, On Bickel and Ritov's conjecture about adaptive estimation of the integral of the square of density derivatives. Ann. Statist. 24 (1996) 682-686. | Zbl 0859.62039
[8] S. Efroïmovich and M. Low, On optimal adaptive estimation of a quadratic functional. Ann. Statist. 24 (1996) 1106-1125. | Zbl 0865.62024
[9] M. Fromont and B. Laurent, Adaptive goodness-of-fit tests in a density model. Technical report. Université Paris 11 (2003). | Zbl 1096.62040
[10] G. Gayraud and K. Tribouley, Wavelet methods to estimate an integrated quadratic functional: Adaptivity and asymptotic law. Statist. Probab. Lett. 44 (1999) 109-122. | Zbl 0947.62029
[11] E. Giné, R. Latala and J. Zinn, Exponential and moment inequalities for
U
-statistics. High Dimensional Probability 2, Progress in Probability 47 (2000) 13-38. | Zbl 0969.60024
[12] W. Hardle, G. Kerkyacharian, D. Picard, A. Tsybakov, Wavelets, Approximations and statistical applications. Lect. Notes Stat. 129 (1998). | MR 1618204 | Zbl 0899.62002
[13] C. Houdré and P. Reynaud-Bouret, Exponential inequalities for U-statistics of order two with constants, in Euroconference on Stochastic inequalities and applications. Barcelona. Birkhauser (2002). | Zbl 1036.60015
[14] I.A. Ibragimov, A. Nemirovski and R.Z. Hasminskii, Some problems on nonparametric estimation in Gaussian white noise. Theory Probab. Appl. 31 (1986) 391-406. | Zbl 0623.62028
[15] I. Johnstone, Chi-square oracle inequalities. State of the art in probability and statistics (Leiden 1999) - IMS Lecture Notes Monogr. Ser., 36. Inst. Math. Statist., Beachwood, OH (1999) 399-418.
[16] B. Laurent, Efficient estimation of integral functionals of a density. Ann. Statist. 24 (1996) 659-681. | Zbl 0859.62038
[17] B. Laurent, Estimation of integral functionals of a density and its derivatives. Bernoulli 3 (1997) 181-211. | Zbl 0872.62044
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Write a polynomial f(x) that satisfies the given conditions. Degree 3
Write a polynomial f(x) that satisfies the given conditions.
Degree 3 polynomial with integer coefficients with zeros −3i and 9/5.
Here we write a polynomial function of 3 with zeros −3i and 9/5.
Let f(x) be the ploynomial function
Since −3i be the zeros of f(x) then +3i be also zeros of f(x)
[because complex conjugate each other]
Since −3i be the zeros of f(x) then (x+3i) be the factor of f(x).
Since +3i be the zeros of f(x) then (x−3i) be tge factor of f(x)
Since 9/5 be the zeros of f(x) , then (x−9/5) be the factor of f(x).
Therefore f(x)=(x+3i)(x−3i)(x−9/6)
=\left({x}^{2}-{\left(3i\right)}^{2}\right)\left(x-\frac{9}{5}\right)
={x}^{2}\left(x-\frac{9}{5}\right)+9\left(x-\frac{9}{5}\right)
={x}^{3}-\frac{9{x}^{2}}{5}+9x-\frac{81}{5}
={x}^{3}-\frac{9{x}^{2}}{5}+9x-\frac{81}{5}
f\left(x\right)={x}^{3}-\frac{9{x}^{2}}{5}+9x-\frac{81}{5}
Zeros -3i and
\frac{9}{5}
Complex zero must have conjugate
x=±3i\text{ }x=\frac{9}{5}
f\left(x\right)=\left(x-3i\right)\left(x+3i\right)\left(x-\frac{9}{5}\right)
=\left({x}^{2}-{\left(3i\right)}^{2}\right)\left(x-\frac{9}{5}\right)
=\left({x}^{2}+9\right)\left(x-\frac{9}{5}\right)
={x}^{3}+9x-\frac{9}{5}{x}^{2}-\frac{81}{5}
={x}^{3}-\frac{9}{5}{x}^{2}+9x-\frac{81}{5}
g\left(x\right)=3-\frac{{x}^{2}}{4}
Consider the following polynomials over
{Z}_{8}
where a is written for [a] in
{Z}_{8}
f\left(x\right)=2{x}^{3}+7x+4,g\left(x\right)=4{x}^{2}+4x+6,h\left(x\right)=6{x}^{2}+3
Find each of the following polynomials with all coefficients in
{Z}_{8}
f(x)g(x)+h(x)
To perform: The operation [43]+[32] in
{Z}_{11}
and indicate the answer in [r] where
0\le r\le m
x\left(4-{x}^{2}\right)\left(2x+1\right)
All the real zeros of the given polynomial are integers. Find the zeros. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)
P\left(x\right)={x}^{3}+5{x}^{2}-x-5
Write the polynomial in factored form.
Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero.
P\left(x\right)=16{x}^{4}-81
How do you write the equation y=-1.7x+8.5 in standard form?
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Time Operator in Quantum Mechanics
Department of Physics, National University Bangladesh, Gazipur, Bangladesh.
We can not only bring time operator in quantum mechanics (non-relativistic) but also determine its Eigen value, commutation relation of its square with energy and some of the properties of time operator like either it is Hermitian or not, either its expectation value is real or complex for a wave packet etc. Exactly these are what I have done.
Quantum Mechanics, Quantum Physics, Quantum Operators, Time Operators
Routh, A. (2019) Time Operator in Quantum Mechanics. Open Access Library Journal, 6, 1-6. doi: 10.4236/oalib.1105816.
Any kind of measurement is organized in space & time. Normally physical quantities are either function of space or function of time and sometimes function of both. In non-relativistic quantum mechanics, though wave function depends on both space and time, only space enters quantum mechanics as operator, time does not. There is no time operator in quantum mechanics. All quantum mechanics textbooks introduce only space as operator, not time. But it is not only possible to produce time operator in quantum mechanics but also we can find out Eigen value of time operator, determines its square’s commutation relation with energy, and prove that it is Hermitian operator; the expectation value of time operator is real for a wave packet and any two wave functions
\Psi \left(r,t\right)
\varphi \left(r,E\right)
are Fourier transform of each other, where
r,t,E
represents respectively position, time and energy of any quantum mechanical particle.
2. Origin of Time Operator
If wave nature of some quantum mechanical free particle is described by de Broglie wave function [1]
\Psi \left(r,t\right)=A{\text{e}}^{i\left(k\cdot r-\omega t\right)}
k
is wave number,
\omega
is angular frequency.
⇒\Psi \left(r,t\right)=A{\text{e}}^{i\frac{\left(p\cdot r-Et\right)}{\hslash }}
[Momentum
p=\hslash k
E=\hslash \omega
⇒\frac{\partial \Psi \left(r,t\right)}{\partial E}=-\frac{i}{\hslash }tA{\text{e}}^{i\frac{\left(p\cdot r-Et\right)}{\hslash }}
⇒\frac{\partial \Psi \left(r,t\right)}{\partial E}=-\frac{i}{\hslash }t\Psi \left(r,t\right)
⇒\frac{\partial }{\partial E}=-\frac{i}{\hslash }t
⇒\stackrel{^}{t}=-\frac{\hslash }{i}\frac{\partial }{\partial E}
⇒\stackrel{^}{t}={i}^{2}\frac{\hslash }{i}\frac{\partial }{\partial E}
⇒\stackrel{^}{t}=i\hslash \frac{\partial }{\partial E}
This is time operator. Though in Zhi-Yong Wang’s paper [2] there was a minus sign in front of “
i\hslash \frac{\partial }{\partial E}
”. We can omit that.
3. Eigen Value of Time Operator
Now we will see how to find out its Eigen value. We have known,
\stackrel{^}{t}=i\hslash \frac{\partial }{\partial E}
⇒\stackrel{^}{t}\Psi =i\hslash \frac{\partial }{\partial E}\Psi
⇒\stackrel{^}{t}\Psi =i\hslash \frac{\partial }{\partial E}\left(A{\text{e}}^{i\frac{\left(p\cdot r-Et\right)}{\hslash }}\right)
⇒\stackrel{^}{t}\Psi =i\hslash A\left(-\frac{i}{\hslash }t\right){\text{e}}^{-iEt}
⇒\stackrel{^}{t}\Psi =-{i}^{2}t\Psi
⇒\stackrel{^}{t}\Psi =t\Psi
⇒\stackrel{^}{t}=t
This is the Eigen value of time operator.
We know that the commutation relation between time operator and energy operator is “
i\hslash
4. Commutation Relation between Square of Time Operator and Energy
Now we will determine the commutation relation between square of time operator and energy.
\left[{\stackrel{^}{t}}^{2},E\right]\Psi =\left({\stackrel{^}{t}}^{2}E-E{\stackrel{^}{t}}^{2}\right)\Psi
⇒\left[{\stackrel{^}{t}}^{2},E\right]\Psi =i\hslash \frac{\partial }{\partial {E}^{2}}\left(E\Psi \right)-i\hslash E\frac{\partial \Psi }{\partial {E}^{2}}
⇒\left[{\stackrel{^}{t}}^{2},E\right]\Psi =i\hslash \left(\frac{\partial }{\partial {E}^{2}}E\right)\Psi +i\hslash E\frac{\partial \Psi }{\partial {E}^{2}}-i\hslash E\frac{\partial \Psi }{\partial {E}^{2}}
⇒\left[{\stackrel{^}{t}}^{2},E\right]\Psi =0
⇒\left[{\stackrel{^}{t}}^{2},E\right]=0
This is the commutation relation between square of time operator and energy and it is zero. This means, square of time operator and energy commutes.
5. Wave Functions Which Depends on Position and Time Is Fourier Transform of Wave Function Dependent on Position and Energy
We can show that two wave functions
\Psi \left(r,t\right)
\varphi \left(r,E\right)
are Fourier transform of each other. If wave function
\Psi \left(r,t\right)
is given by that,
\Psi \left(r,t\right)=A\Psi \left(r\right){\text{e}}^{-\frac{iEt}{\hslash }}
⇒{\Psi }_{E}\left(t\right)=A{\text{e}}^{-\frac{iEt}{\hslash }}
⇒〈t|{\Psi }_{E}〉=A{\text{e}}^{-\frac{iEt}{\hslash }}
⇒〈{\Psi }_{E}|{t}^{\prime }〉={A}^{\ast }{\text{e}}^{-\frac{iE{t}^{\prime }}{\hslash }}
By multiplying Equations (4) and (5),
〈t|{\Psi }_{E}〉〈{\Psi }_{E}|{t}^{\prime }〉={|A|}^{2}{\text{e}}^{\frac{iE\left({t}^{\prime }-t\right)}{\hslash }}
⇒{|A|}^{2}\int {\text{e}}^{\frac{iE\left({t}^{\prime }-t\right)}{\hslash }}\text{d}E=\int 〈t|{\Psi }_{E}〉〈{\Psi }_{E}|{t}^{\prime }〉\text{d}E
⇒{|A|}^{2}\int {\text{e}}^{\frac{iE\left({t}^{\prime }-t\right)}{\hslash }}\text{d}E=\int 〈t|E〉〈E|{t}^{\prime }〉\text{d}E
⇒{|A|}^{2}\int {\text{e}}^{\frac{iE\left({t}^{\prime }-t\right)}{\hslash }}\text{d}E=〈t|\int E〉〈E|{t}^{\prime }〉\text{d}E
⇒{|A|}^{2}\int {\text{e}}^{\frac{iE\left({t}^{\prime }-t\right)}{\hslash }}\text{d}E=〈t|{t}^{\prime }〉
⇒{|A|}^{2}\int {\text{e}}^{\frac{iE\left({t}^{\prime }-t\right)}{\hslash }}\text{d}E=\delta \left(t-{t}^{\prime }\right)
⇒{|A|}^{2}\int {\text{e}}^{\frac{iE\left({t}^{\prime }-t\right)}{\hslash }}\text{d}E=\frac{1}{2\pi }\int {\text{e}}^{\left(t-{t}^{\prime }\right)}\text{d}\omega
⇒{|A|}^{2}\int {\text{e}}^{\frac{iE\left({t}^{\prime }-t\right)}{\hslash }}\text{d}E=\frac{1}{2\pi \hslash }\int {\text{e}}^{\left(t-{t}^{\prime }\right)}\text{d}\left(\hslash \omega \right)
In the left hand side, We interchange t & t' and get
⇒{|A|}^{2}\int {\text{e}}^{\frac{iE\left(t-{t}^{\prime }\right)}{\hslash }}\text{d}E=\frac{1}{2\pi \hslash }\int {\text{e}}^{iE\left(t-{t}^{\prime }\right)}\text{d}E
⇒{|A|}^{2}=\frac{1}{2\pi \hslash }
⇒A=\frac{1}{\sqrt{2\pi \hslash }}
Substituting the value of A into Equation (3),
⇒〈t|{\Psi }_{E}〉=\frac{1}{\sqrt{2\pi \hslash }}{\text{e}}^{-\frac{iEt}{\hslash }}
⇒〈t|E〉=\frac{1}{\sqrt{2\pi \hslash }}{\text{e}}^{-\frac{iEt}{\hslash }}
⇒〈E|t〉={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}{\text{e}}^{\frac{iEt}{\hslash }}
〈t|\alpha 〉=\int {\text{d}}^{3}E〈t|E〉〈E|\alpha 〉
⇒〈t|\alpha 〉=\int {\text{d}}^{3}E{\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}{\text{e}}^{-\frac{iEt}{\hslash }}〈E|\alpha 〉
⇒〈t|\alpha 〉={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}\int 〈E|\alpha 〉{\text{e}}^{-\frac{iEt}{\hslash }}{\text{d}}^{3}E
⇒〈t|{\Psi }_{\alpha }〉={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}\int 〈E|{\phi }_{\alpha }〉{\text{e}}^{-\frac{iEt}{\hslash }}{\text{d}}^{3}E
⇒{\Psi }_{\alpha }\left(t\right)={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}\int {\phi }_{\alpha }\left(E\right){\text{e}}^{-\frac{iEt}{\hslash }}{\text{d}}^{3}E
〈E|\alpha 〉=\int {\text{d}}^{3}t〈E|t〉〈t|\alpha 〉
⇒〈E|\alpha 〉={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}\int {\text{e}}^{\frac{iEt}{\hslash }}{\text{d}}^{3}t〈t|\alpha 〉
⇒〈E|\alpha 〉={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}〈t|\alpha 〉\int {\text{e}}^{\frac{iEt}{\hslash }}{\text{d}}^{3}t
⇒〈E|{\phi }_{\alpha }〉={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}〈t|{\Psi }_{\alpha }〉\int {\text{e}}^{\frac{iEt}{\hslash }}{\text{d}}^{3}t
⇒{\phi }_{\alpha }\left(E\right)={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}{\Psi }_{\alpha }\left(t\right)\int {\text{e}}^{\frac{iEt}{\hslash }}{\text{d}}^{3}t
We can generalize (8) & (9) respectively and write these equations,
\Psi \left(t\right)={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}\int \phi \left(E\right){\text{e}}^{-\frac{iEt}{\hslash }}{\text{d}}^{3}E
\phi \left(E\right)={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}\int \Psi \left(t\right){\text{e}}^{\frac{iEt}{\hslash }}{\text{d}}^{3}t
We can rewrite Equation (3) into this form,
\Psi \left(r,t\right)=\Psi \left(t\right){\text{e}}^{i\frac{p\cdot r}{\hslash }}
⇒\Psi \left(r,t\right)={\text{e}}^{i\frac{p\cdot r}{\hslash }}{\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}\int \phi \left(E\right){\text{e}}^{-\frac{iEt}{\hslash }}{\text{d}}^{3}E
[from (10)]
⇒\Psi \left(r,t\right)={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}\int \phi \left(E\right){\text{e}}^{i\frac{p\cdot r}{\hslash }}{\text{e}}^{-\frac{iEt}{\hslash }}{\text{d}}^{3}E
⇒\Psi \left(r,t\right)={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}\int \phi \left(E,r\right){\text{e}}^{-\frac{iEt}{\hslash }}{\text{d}}^{3}E
\phi \left(E,r\right)=\phi \left(E\right){\text{e}}^{i\frac{p\cdot r}{\hslash }}
Like Equation (12) we can write energy dependent wave function like this,
\phi \left(r,E\right)=\phi \left(E\right){\text{e}}^{i\frac{p\cdot r}{\hslash }}
⇒\phi \left(r,E\right)={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}\int \Psi \left(t\right){\text{e}}^{\frac{iEt}{\hslash }}{\text{d}}^{3}t\text{ }{\text{e}}^{i\frac{p\cdot r}{\hslash }}
⇒\phi \left(r,E\right)={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}\int \Psi \left(t\right){\text{e}}^{i\frac{p\cdot r}{\hslash }}{\text{e}}^{\frac{iEt}{\hslash }}{\text{d}}^{3}t
⇒\phi \left(r,E\right)={\left(\sqrt{2\pi \hslash }\right)}^{-\frac{1}{2}}\int \Psi \left(r,t\right){\text{e}}^{\frac{iEt}{\hslash }}{\text{d}}^{3}t
\Psi \left(r,t\right)=\Psi \left(t\right){\text{e}}^{i\frac{p\cdot r}{\hslash }}
From Equations (13) & (14) we can see that wave function.
\Psi \left(r,t\right)
\phi \left(r,E\right)
are each other’s Fourier transform. Until now, wave functions only depend on position, time, and momentum. For the first time, I have showed that wave also depend on energy.
6. Expectation Value of Time Operator
Now we will see that the expectation value of time operator is real.
〈t〉={\int }_{-\infty }^{\infty }{\Psi }^{\ast }t\Psi \text{d}E
⇒〈t〉=i\hslash {\int }_{-\infty }^{\infty }{\Psi }^{\ast }\frac{\partial }{\partial E}\Psi \text{d}E
⇒〈t〉=i\hslash {\left[{\Psi }^{\ast }\int \frac{\partial \Psi }{\partial E}\text{d}E\right]}_{-\infty }^{\infty }-i\hslash {\int }_{-\infty }^{\infty }\left(\frac{\partial {\Psi }^{\ast }}{\partial E}\int \frac{\partial \Psi }{\partial E}\text{d}E\right)\text{d}E
⇒〈t〉=0-i\hslash {\int }_{-\infty }^{\infty }\left(\frac{\partial {\Psi }^{\ast }}{\partial E}\Psi \right)\text{d}E
⇒〈t〉={\int }_{-\infty }^{\infty }\Psi {\left(i\hslash \frac{\partial {\Psi }^{\ast }}{\partial E}\right)}^{\ast }\text{d}E
⇒〈t〉={\int }_{-\infty }^{\infty }\Psi t{\Psi }^{\ast }\text{d}E
⇒〈t〉={\int }_{-\infty }^{\infty }{\left({\Psi }^{\ast }t\Psi \right)}^{\ast }\text{d}E
⇒〈t〉={〈t〉}^{\ast }
This is possible only when time operator’s expectation value is real number. So
〈t〉
is real. In other papers [2] [3], it has proved that time operator is a Hermitian operator. If any operator is Hermitian then its expected value is real. I don’t prove that time operator’s self-adjointness but directly prove that time operator’s expectation value is real.
Definition of time operator and its commutation relation with energy are given in many papers [2] [3], but we have shown that its square commutes with energy;
\Psi \left(r,t\right)
\phi \left(r,E\right)
are each other’s Fourier transform and expectation value of time operator is real. We also have determined its Eigen value. Readers of this paper can use time operator and create a new kind of equation of motion which will become alternate of Schrodinger equation.
[1] Zettili, N. (2003) Quantum Mechanics: Concepts and Applications. American Journal of Physics, 71, 93. https://doi.org/10.1119/1.1522702
[2] Wang, Z.-Y. and Xiong, C.-D. (2007) How to Introduce Time Operator. Annals of Physics, 322, 2304-2314. https://doi.org/10.1016/j.aop.2006.10.007
[3] Olkhovsky, V., Recami, E. and Gerasimchuk, A. (1974) Time Operator in Quantum Mechanics. Il Nuovo Cimento A (1965-1970), 22, 263-278.
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m (Added link to DG's Imagery Product Samples page)
m (→HPFA based sharpening: Link to external custom add-on implementing the HPFA image fusion technique (temporary))
{\displaystyle {\frac {W}{m^{2}*sr*nm}}}
{\displaystyle L_{\lambda {\text{Pixel, Band}}}={\frac {K_{\text{Band}}*q_{\text{Pixel, Band}}}{\Delta \lambda _{\text{Band}}}}}
{\displaystyle L_{\lambda {\text{Pixel,Band}}}}
{\displaystyle K_{\text{Band}}}
{\displaystyle q_{\text{Pixel,Band}}}
{\displaystyle \Delta _{\lambda _{\text{Band}}}}
{\displaystyle \rho _{p}={\frac {\pi *L\lambda *d^{2}}{ESUN\lambda *cos(\Theta _{S})}}}
{\displaystyle \rho }
{\displaystyle \pi }
{\displaystyle L\lambda }
{\displaystyle d}
{\displaystyle Esun}
{\displaystyle cos(\theta _{s})}
{\displaystyle {\frac {W}{m^{2}*\mu m}}}
The following screenshots exemplify the High Pass Filtering Addition fusion technique applied in a fragment of the QuickBird acquisition "04APR05050541-X2AS_R1C1-000000186011_01_P001-Sri_Lanka-Kokilai_Lagoon" which is publicly available via the GLCF. Currently, an unofficial implementation of this technique for GRASS-GIS is available as a custom grass-addon called i.fusion.hpf
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Deng, Dianliang
In the present paper, by using the inequality due to Talagrand's isoperimetric method, several versions of the bounded law of iterated logarithm for a sequence of independent Banach space valued random variables are developed and the upper limits for the non-random constant are given.
Classification : 60F05, 60B12, 60F99
Mots clés : Banach space, bounded law of iterated logarithm, isoperimetric inequality, Rademacher series, self-normalizer
author = {Deng, Dianliang},
title = {On the bounded laws of iterated logarithm in {Banach} space},
AU - Deng, Dianliang
TI - On the bounded laws of iterated logarithm in Banach space
Deng, Dianliang. On the bounded laws of iterated logarithm in Banach space. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 19-37. doi : 10.1051/ps:2005002. http://www.numdam.org/articles/10.1051/ps:2005002/
[1] A. De Acosta, Inequalities for
B
-valued random variables with application to the law of large numbers. Ann. Probab. 9 (1981) 157-161. | Zbl 0449.60002
[2] B. Von Bahr and C. Esseen, Inequalities for the
r
th absolute moments of a sum of random variables,
1\le r\le 2
[3] X. Chen, On the law of iterated logarithm for independent Banach space valued random variables. Ann. Probab. 21 (1993) 1991-2011. | Zbl 0791.60005
[4] X. Chen, The Kolmogorov’s LIL of
B
-valued random elements and empirical processes. Acta Mathematica Sinica 36 (1993) 600-619. | Zbl 0785.60019
[5] Y.S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martigales. Springer-Verlag, New York (1978). | MR 513230 | Zbl 0399.60001
[6] D. Deng, On the Self-normalized Bounded Laws of Iterated Logarithm in Banach Space. Stat. Prob. Lett. 19 (2003) 277-286. | Zbl 1113.60300
[7] U. Einmahl, Toward a general law of the iterated logarithm in Banach space. Ann. Probab. 21 (1993) 2012-2045. | Zbl 0790.60034
[8] E. Gine and J. Zinn, Some limit theorem for emperical processes. Ann. Probab. 12 (1984) 929-989. | Zbl 0553.60037
[9] A. Godbole, Self-normalized bounded laws of the iterated logarithm in Banach spaces, in Probability in Banach Spaces 8, R. Dudley, M. Hahn and J. Kuelbs Eds. Birkhäuser Progr. Probab. 30 (1992) 292-303. | Zbl 0787.60011
[10] P. Griffin and J. Kuelbs, Self-normalized laws of the iterated logarithm. Ann. Probab. 17 (1989) 1571-1601. | Zbl 0687.60033
[11] P. Griffin and J. Kuelbs, Some extensions of the LIL via self-normalizations. Ann. Probab. 19 (1991) 380-395. | Zbl 0722.60028
[12] M. Ledoux and M. Talagrand, Characterization of the law of the iterated logarithm in Babach spaces. Ann. Probab. 16 (1988) 1242-1264. | Zbl 0662.60008
[13] M. Ledoux and M. Talagrand, Some applications of isoperimetric methods to strong limit theorems for sums of independent random variables. Ann. Probab. 18 (1990) 754-789. | Zbl 0713.60005
[14] M. Ledoux and M. Talagrand, Probability in Banach Space. Springer-Verlag, Berlin (1991). | MR 1102015 | Zbl 0748.60004
[15] R. Wittmann, A general law of iterated logarithm. Z. Wahrsch. verw. Gebiete 68 (1985) 521-543. | Zbl 0547.60036
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Algebraic Loop Concepts - MATLAB & Simulink - MathWorks Benelux
How the Algebraic Loop Solver Works
Trust-Region and Line-Search Algorithms in the Algebraic Loop Solver
Limitations of the Algebraic Loop Solver
Implications of Algebraic Loops in a Model
In a Simulink® model, an algebraic loop occurs when a signal loop exists with only direct feedthrough blocks within the loop. Direct feedthrough means that Simulink needs the value of the block’s input signal to compute its output at the current time step. Such a signal loop creates a circular dependency of block outputs and inputs in the same time-step. This results in an algebraic equation that needs solving at each time-step, adding computational cost to the simulation.
Some examples of blocks with direct feedthrough inputs are:
State-Space, when the D matrix coefficient is nonzero
Transfer Fcn, when the numerator and denominator are of the same order
Zero-Pole, when the block has as many zeros as poles
Nondirect feedthrough blocks maintain a State variable. Two examples are Integrator and Unit Delay.
To determine if a block has direct feedthrough, read the Characteristics section of the block reference page.
The figure shows an example of an algebraic loop. The Sum block is an algebraic variable xa that is constrained to equal the first input u minus xa (for example, xa = u – xa).
The solution of this simple loop is xa = u/2.
Simulink contains a suite of numerical solvers for simulating ordinary differential equations (ODEs), which are systems of equations that you can write as
\stackrel{˙}{x}=f\left(x,t\right),
where x is the state vector and t is the independent time variable.
Some systems of equations contain additional constraints that involve the independent variable and the state vector, but not the derivative of the state vector. Such systems are called differential algebraic equations (DAEs),
The term algebraic refers to equations that do not involve any derivatives. You can express DAEs that arise in engineering in the semi-explicit form
\begin{array}{l}\stackrel{˙}{x}=f\left(x,{x}_{a},t\right)\\ 0=g\left(x,{x}_{a},t\right),\end{array}
f and g can be vector functions.
The first equation is the differential equation.
The second equation is the algebraic equation.
The vector of differential variables is x.
The vector of algebraic variables is xa.
In Simulink models, algebraic loops are algebraic constraints. Models with algebraic loops define a system of differential algebraic equations. Simulink solves the algebraic equations (the algebraic loop) numerically for xa at each step of the ODE solver.
The model in the figure is equivalent to this system of equations in semi-explicit form:
\begin{array}{l}\stackrel{˙}{x}=f\left(x,{x}_{a},t\right)={x}_{a}\\ 0=g\left(x,{x}_{a},t\right)=-x+u-2{x}_{a}.\end{array}
At each step of the ODE solver, the algebraic loop solver must solve the algebraic constraint for xa before calculating the derivative
\stackrel{˙}{x}
Algebraic constraints:
Occur when modeling physical systems, often due to conservation laws, such as conservation of mass and energy
Occur when you choose a particular coordinate system for a model
Help impose design constraints on system responses in a dynamic system
Use Simscape™ to model systems that span mechanical, electrical, hydraulic, and other physical domains as physical networks. Simscape constructs the DAEs that characterize the behavior of a model. The software integrates these equations with the rest of the model and then solves the DAEs directly. Simulink solves the variables for the components in the different physical domains simultaneously, avoiding problems with algebraic loops.
When a model contains an algebraic loop, Simulink uses a nonlinear solver at each time step to solve the algebraic loop. The solver performs iterations to determine the solution to the algebraic constraint, if there is one. As a result, models with algebraic loops can run more slowly than models without algebraic loops.
Simulink uses a dogleg trust region algorithm to solve algebraic loops. The tolerance used is smaller than the ODE solver Reltol and Abstol. This is because Simulink uses the “explicit ODE method” to solve Index-1 differential algebraic equations (DAEs).
For the algebraic loop solver to work,
There must be one block where the loop solver can break the loop and attempt to solve the loop.
The model should have real double signals.
The underlying algebraic constraint must be a smooth function
For example, suppose your model has a Sum block with two inputs—one additive, the other subtractive. If you feed the output of the Sum block to one of the inputs, you create an algebraic loop where all of the blocks include direct feedthrough.
The Sum block cannot compute the output without knowing the input. Simulink detects the algebraic loop, and the algebraic loop solver solves the loop using an iterative loop. In the Sum block example, the software computes the correct result this way:
xa(t) = u(t) / 2. (1)
The algebraic loop solver uses a gradient-based search method, which requires continuous first derivatives of the algebraic constraint that correspond to the algebraic loop. As a result, if the algebraic loop contains discontinuities, the algebraic loop solver can fail.
For more information, see Solving Index-1 DAEs in MATLAB and Simulink 1
The Simulink algebraic loop solver uses one of two algorithms to solve algebraic loops:
By default, Simulink chooses the best algebraic loop solver and may switch between the two methods during simulation. To explicitly enable automatic algebraic loop solver selection for your model, at the MATLAB® command line, enter:
To switch to the trust-region algorithm, at the MATLAB command line, enter:
If the algebraic loop solver cannot solve the algebraic loop with the trust-region algorithm, try simulating the model using the line-search algorithm.
To switch to the line-search algorithm, at the MATLAB command line, enter:
Shampine and Reichelt’s nleqn.m code
The Fortran program HYBRD1 in the User Guide for MINPACK-1 2
Powell’s “A Fortran subroutine for solving systems in nonlinear equations,” in Numerical Methods for Nonlinear Algebraic Equations3
Trust-Region Methods for Nonlinear Minimization (Optimization Toolbox).
Line Search (Optimization Toolbox).
Algebraic loop solving is an iterative process. The Simulink algebraic loop solver is successful only if the algebraic loop converges to a definite answer. When the loop fails to converge, or converges too slowly, the simulation exits with an error.
The algebraic loop solver cannot solve algebraic loops that contain any of the following:
Blocks with discrete-valued outputs
Blocks with nondouble or complex outputs
If your model contains an algebraic loop:
You cannot generate code for the model.
The Simulink algebraic loop solver might not be able to solve the algebraic loop.
While Simulink is trying to solve the algebraic loop, the simulation can execute slowly.
For most models, the algebraic loop solver is computationally expensive for the first time step. Simulink solves subsequent time steps rapidly because a good starting point for xa is available from the previous time step.
Compare Solvers | Zero-Crossing Detection | Algebraic Constraint | Descriptor State-Space
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DISCUSS DISCOVER: How Many Real Zeros Can a Polynomial Have?
DISCUSS DISCOVER: How Many Real Zeros Can a Polynomial Have? Give examples of po
DISCUSS DISCOVER: How Many Real Zeros Can a Polynomial Have? Give examples of polynomials that have the following properties, or explain why it is impossible to find such a polynomial.
(a) A polynomial of degree 3 that has no real zeros
(b) A polynomial of degree 4 that has no real zeros
(c) A polynomial of degree 3 that has three real zeros, only one of which is rational
(d) A polynomial of degree 4 that has four real zeros, none of which is rational
What must be true about the degree of a polynomial with integer coefficients if it has no real zeros?
(a). Let a 3rd polynomial function P(x).
If the leading term is positive the end behavior is :
y\to \mathrm{\infty }
x\to -\mathrm{\infty }
y\to +\mathrm{\infty }
x\to +\mathrm{\infty }
If the leading term is negative the end behavior is :
y\to +\mathrm{\infty }
x\to -\mathrm{\infty }
y\to -\mathrm{\infty }
x\to +\mathrm{\infty }
As we see in each case the end behavior is different as
x\to -\mathrm{\infty }
x\to +\mathrm{\infty }
which means that the graph of a third degree polynomial must cross the x-axis at least once.
This means that every polynomial P(x) of degree 3 has at least one real zero.
(b). Let a 4th degree polynomial P(x) .
P\left(x\right)={x}^{4}+{x}^{2}+3
{x}^{4}
{x}^{2}
are non negative as even powers of x and the constant term is positive.
P(x) is strictly positive. So the polynomial P(x) has no real zeros.
(c). Let the 3rd degree polynomial
P\left(x\right)=\left(x-\sqrt{2}\right)\left(x-\pi \right)\left(x-1\right)
P\left(x\right)=\left({x}^{2}-x\pi -\sqrt{2}x-\sqrt{2}\pi \right)\left(x-1\right)
={x}^{3}-{x}^{2}-\pi {x}^{2}+\pi x-\sqrt{2}{x}^{2}+\sqrt{2}x+\sqrt{2}\pi x-\sqrt{2}\pi
={x}^{3}-\left(1+\pi +\sqrt{2}\right){x}^{2}+\left(\pi +\sqrt{2}+\sqrt{2}\pi \right)x-\sqrt{2}\pi
A 3rd degree polynomial with one rational zero
x=1
and twp irrational zeros
x=\sqrt{2},\pi
By our guidelines, we are supposed to answer first three questions only.
\frac{1}{3},\frac{2}{9},\frac{3}{27},\frac{4}{81},...
3,8,13,18,...,48
2{x}^{3}-3{x}^{2}+{x}^{5}
. Which of the following statements about the polynomial are not correct?
A. The standard form of the polynomial is
{x}^{5}+2{x}^{3}-3{x}^{2}
B. The polynomial is a trinomial.
C. The polynomial is quintic.
D. The polynomial has a linear term.
g\left(x\right)=\frac{{x}^{3}}{2}-{x}^{2}+2
x-y=3
{x}^{3}-{y}^{3}-9xy
Expand u(u+2).
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Gross National Product (GNP) Deflator
What Is the Gross National Product (GNP) Deflator?
The gross national product deflator is an economic metric that accounts for the effects of inflation in the current year's gross national product (GNP) by converting its output to a level relative to a base period.
The GNP deflator can be confused with the more commonly used gross domestic product (GDP) deflator. The GDP deflator uses the same equation as the GNP deflator, but with nominal and real GDP rather than GNP.
The GNP deflator provides an alternative to the Consumer Price Index (CPI) and can be used in conjunction with it to analyze some changes in trade flows and the effects on the welfare of people within a relatively open market country.
The higher the GNP deflator, the higher the rate of inflation for the period.
Understanding the Gross National Product (GNP) Deflator
The GNP deflator is simply the adjustment for inflation that is made to nominal GNP to produce real GNP. The GNP deflator provides an alternative to the Consumer Price Index (CPI) and can be used in conjunction with it to analyze some changes in trade flows and the effects on the welfare of people within a relatively open market country.
The CPI is based upon a basket of goods and services, while the GNP deflator incorporates all of the final goods produced by an economy. This allows the GNP deflator to more accurately capture the effects of inflation since it's not limited to a smaller subset of goods.
Calculating the Gross National Product (GNP) Deflator
The GNP deflator is calculated with the following formula:
\text{GNP Deflator}\ = \ \left(\frac{\text{Nominal GNP}}{\text{Real GNP}}\right)\times 100
GNP Deflator = (Real GNPNominal GNP)×100
The result is expressed as a percentage, usually with three decimal places.
The first step to calculating the GNP deflator is to determine the base period for analysis. In theory, you can work with GDP and foreign earnings data for the base period and current periods, and then extract the figures needed for the deflator calculation. However, nominal GNP and real GNP figures, as well as the deflator charted over time, can usually be accessed through releases from central banks or other economic entities.
In the United States, the Bureau of Economic Analysis (BEA), the St. Louis Federal Reserve Bank, and others provide this data, as well as other indicators that track similar economic statistics that measure essentially the same thing but through different formulations. So actually calculating the GNP deflator is usually unnecessary. The more important task is how to interpret the data that the GNP deflator is applied to.
Interpreting GNP Figures
The GNP deflator, as mentioned, is just the inflation adjustment. The higher the GNP deflator, the higher the rate of inflation for the period. The relevant question is what having an inflation-adjusted gross national product—the real GNP—actually tells you.
The real GNP is simply the actual national income of the country being measured. It doesn't care where the production is located in the world as long as the earnings come back home.
In terms of differences between real GNP and real GDP, real GDP is the preferred measure of U.S. economic health. Real GNP shows how the U.S. is doing in terms of its foreign investments in addition to domestic production.
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Lewis Structure | Brilliant Math & Science Wiki
Sravanth C., Damon Bernard, Skanda Prasad, and
Ready 4 Races
Gilbert N. Lewis introduced a diagrammatic system to represent the valence shell electrons in an atom. It helps us a lot while doing chemical bonding. He used dots to represent the valence electrons. He used lines to indicate bond between two electrons. Here's an example of carbon, with its valence electrons using Lewis dot structure.
Calculating Number of Lone Pairs and Bonds
Features of Lewis Structures
Gilbert Lewis represented the valence electrons using dots. As in the example above, you can see the valence electrons of carbon being represented as dots.
But, when we have to represent a chemical bond between two different atoms, we normally use a different symbol for each atom taking part in the reaction. Let's see an example of a molecule involving two different atoms.
In this compound (carbon dioxide), we find that the electrons of oxygen are represented by dots whereas valence electrons of carbon are represented by a cross.
Note: In case the bond is between two atoms of the same kind, there is no need to use different symbols.
We know that the atoms tend to attain the octet state (or duplet state in case of hydrogen) and attain stability. For attaining stability, atoms combine with other atoms, or with the same type of atom. To show the bondings diagrammatically, we need to get to know about the lone pair and bonded pair. As said earlier, (generally) when we represent a chemical bond we draw a line instead of keeping the bonded electrons as such.
Sometimes, not all electrons take part in forming a bond; they are kept away from the bonded pair but are kept around the atom. These un-bonded electrons are generally kept in pairs. Such electrons are called lone pairs.
Bonded pair: There are the pair(s) of electrons which take part in forming a bond. They are normally represented by lines to avoid confusion between them and the lone pairs.
Here's an example of
\text{NH}_3
(ammonia), where you can observe the lone pair and the bonded pair:
As said earlier, the bonded pairs are shown as lines. But it's not compulsory to do so; you can keep the bonded pairs as it is. As in the example above, the bonded pairs are not shown as lines. But generally we use lines as it keeps things neat. Here's an example of
\text{H}_2\text{O}
(water):
We can observe that the two pairs of electrons have disappeared and in their place a single line is placed, showing a bonded pair. The number of lines shows the number of pairs of bonded electrons.
For a neutral atom, we see that the following equation holds good:
(\text{no. of bonds}) = (\text{total electrons in a complete valence shell}) - (\text{no. of valence electrons excluding the bonded pair}).
Suppose we're given the exercise to find the amount of bonds and lone pairs in the molecule
\text{NH}_3
. First let's find the amount of total electrons needed for the noble gas configuration. This will be given by
\sum _{ j=0 }^{k}\alpha_{j}\cdot \beta_{j}
\alpha_{j}
is a constant that will follow from the octet rule. For hydrogen atoms,
\alpha_{j}
is equal to 2. For any other atom,
\alpha_{j}
\beta_{j}
is defined as the number of atoms per kind. Let's clarify that a bit by applying it. There is 1
\text{N}
atom. Therefore
\alpha_{0} = 8
, eight electrons needed per atom, and
\beta_{0} = 1
, one nitrogen atom. There are 3
\text{H}
atoms. Therefore
\alpha_{1} = 2
, two electrons needed per atom. And
\beta_{1} = 3
, three hydrogen atoms.
\text{NH}_3
\sum_{ j=0 }^{k}\alpha_{j}\cdot \beta_{j} = 8\cdot1 + 2\cdot3 = 14
14 electrons are necessary to obtain the noble gas configuration. Each pair contains two electrons, so there are
\frac{14}{2}=\boxed{7}
bonds and lone pairs needed for
\text{NH}_3
Now let's take a look at the amount of valence electrons in
\text{NH}_3
\text{N}
has 5 valence electrons in its outer shell. Which means that for "all" nitrogen atoms we have
1\cdot5=5
valence electrons in total.
\text{H}
has 1 valence electron in its outer shell. Which means that for all hydrogen atoms we have
3\cdot1=3
valence electrons. And therefore the total amount of valence electrons is
1\cdot5 + 3\cdot1 = 8
valence electrons. This comes down to
\frac{8}{2} = \boxed{4}
pairs available (since we're talking about valence electrons in the outer shell).
(\text{no. of bonds}) = 7 - 4 = 3
And when we check we see that, indeed,
\text{NH}_3
has three bonds. The amount of remaining pairs are the lone pairs. They can be calculated by subtracting the no. of bonds from the available bonds. E.g.
4 - 3 = 1
. So there is one lone pair. And as we can see, that is also true.
Each bond is the result of formation of a bond which means sharing of electron pair between a minimum of two atoms.
Each bonded atom contributes at least one electron for sharing, as we can see in all of the above images of Lewis dot structures.
The atoms which share electrons do so because of the need to satisfy octet configuration (In few cases, duplet configuration).
If only one electron pair is shared, the atoms are said to be bonded by a single covalent bond.
Similarly. if two electron pairs are shared, the atoms are said to be bonded by a double covalent bond.
The maximum electron pairs that can be shared is three. If that is the case, the atoms are said to be bonded by a triple covalent bond.
Cite as: Lewis Structure. Brilliant.org. Retrieved from https://brilliant.org/wiki/lewis-structure/
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m (→Automatising Conversions: Added link to https://github.com/NikosAlexandris/i.quickbird.toar)
m (→Deriving Physical Quantities: Added reference to source for band parameters)
Converting Digital Numbers to Radiance/Reflectance requires knowledge about the sensor's specific spectral band parameters. These are, as extracted from the document ''Radiometric Use of QuickBird Imagery, Technical Note. 2005-11-07, by Keith Krause.''
{\displaystyle {\frac {W}{m^{2}*sr*nm}}}
{\displaystyle L_{\lambda {\text{Pixel, Band}}}={\frac {K_{\text{Band}}*q_{\text{Pixel, Band}}}{\Delta \lambda _{\text{Band}}}}}
{\displaystyle L_{\lambda {\text{Pixel,Band}}}}
{\displaystyle K_{\text{Band}}}
{\displaystyle q_{\text{Pixel,Band}}}
{\displaystyle \Delta _{\lambda _{\text{Band}}}}
{\displaystyle \rho _{p}={\frac {\pi *L\lambda *d^{2}}{ESUN\lambda *cos(\Theta _{S})}}}
{\displaystyle \rho }
{\displaystyle \pi }
{\displaystyle L\lambda }
{\displaystyle d}
{\displaystyle Esun}
{\displaystyle cos(\theta _{s})}
{\displaystyle {\frac {W}{m^{2}*\mu m}}}
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Dedecker, Jérôme ; Louhichi, Sana
We continue the investigation started in a previous paper, on weak convergence to infinitely divisible distributions with finite variance. In the present paper, we study this problem for some weakly dependent random variables, including in particular associated sequences. We obtain minimal conditions expressed in terms of individual random variables. As in the i.i.d. case, we describe the convergence to the gaussian and the purely non-gaussian parts of the infinitely divisible limit. We also discuss the rate of Poisson convergence and emphasize the special case of Bernoulli random variables. The proofs are mainly based on Lindeberg's method.
Mots clés : infinitely divisible distributions, Lévy processes, weak dependence, association, binary random variables, number of exceedances
author = {Dedecker, J\'er\^ome and Louhichi, Sana},
title = {Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences},
AU - Louhichi, Sana
TI - Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences
Dedecker, Jérôme; Louhichi, Sana. Convergence to infinitely divisible distributions with finite variance for some weakly dependent sequences. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 38-73. doi : 10.1051/ps:2005003. http://www.numdam.org/articles/10.1051/ps:2005003/
[1] A. Araujo and E. Giné, The central limit theorem for real and Banach space valued random variables. Wiley, New York (1980). | MR 576407
[2] A.D. Barbour, L. Holst and S. Janson, Poisson approximation. Clarendon Press, Oxford (1992). | MR 1163825 | Zbl 0746.60002
[3] R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life: Probability Models. Silver Spring, MD (1981).
[4] T. Birkel, On the convergence rate in the central limit theorem for associated processes. Ann. Probab. 16 (1988) 1685-1698. | Zbl 0658.60039
[5] A.V. Bulinski, On the convergence rates in the CLT for positively and negatively dependent random fields, in Probability Theory and Mathematical Statistics, I.A. Ibragimov and A. Yu. Zaitsev Eds. Gordon and Breach Publishers, Singapore, (1996) 3-14. | Zbl 0873.60011
[6] L.H.Y. Chen, Poisson approximation for dependent trials. Ann. Probab. 3 (1975) 534-545. | Zbl 0335.60016
[7] J.T. Cox and G. Grimmett, Central limit theorems for associated random variables and the percolation models. Ann. Probab. 12 (1984) 514-528. | Zbl 0536.60094
[8] J. Dedecker and S. Louhichi, Conditional convergence to infinitely divisible distributions with finite variance. Stochastic Proc. Appl. (To appear.) | MR 2132596 | Zbl 1070.60033
[9] P. Doukhan and S. Louhichi, A new weak dependence condition and applications to moment inequalities. Stochastic Proc. Appl. 84 (1999) 313-342. | Zbl 0996.60020
[10] J. Esary, F. Proschan and D. Walkup, Association of random variables with applications. Ann. Math. Statist. 38 (1967) 1466-1476. | Zbl 0183.21502
[11] C. Fortuin, P. Kastelyn and J. Ginibre, Correlation inequalities on some ordered sets. Comm. Math. Phys. 22 (1971) 89-103. | Zbl 0346.06011
[12] B.V. Gnedenko and A.N. Kolmogorov, Limit distributions for sums of independent random variables. Addison-Wesley Publishing Company (1954). | MR 62975 | Zbl 0056.36001
[13] L. Holst and S. Janson, Poisson approximation using the Stein-Chen method and coupling: number of exceedances of Gaussian random variables. Ann. Probab. 18 (1990) 713-723. | Zbl 0713.60047
[14] T. Hsing, J. Hüsler and M.R. Leadbetter, On the Exceedance Point Process for a Stationary Sequence. Probab. Theory Related Fields 78 (1988) 97-112. | Zbl 0619.60054
[15] W.N. Hudson, H.G. Tucker and J.A Veeh, Limit distributions of sums of m-dependent Bernoulli random variables. Probab. Theory Related Fields 82 (1989) 9-17. | Zbl 0672.60033
[16] A. Jakubowski, Minimal conditions in
p
-stable limit theorems. Stochastic Proc. Appl. 44 (1993) 291-327. | Zbl 0771.60015
p
-stable limit theorems -II. Stochastic Proc. Appl. 68 (1997) 1-20. | Zbl 0890.60024
[18] K. Joag-Dev and F. Proschan, Negative association of random variables, with applications. Ann. Statist. 11 (1982) 286-295. | Zbl 0508.62041
[19] O. Kallenberg, Random Measures. Akademie-Verlag, Berlin (1975). | MR 431372 | Zbl 0345.60031
[20] M. Kobus, Generalized Poisson Distributions as Limits of Sums for Arrays of Dependent Random Vectors. J. Multi. Analysis (1995) 199-244. | Zbl 0821.60032
[21] M.R Leadbetter, G. Lindgren and H. Rootzén, Extremes and related properties of random sequences and processes. New York, Springer (1983). | MR 691492 | Zbl 0518.60021
[22] C.M. Newman, Asymptotic independence and limit theorems for positively and negatively dependent random variables, in Inequalities in Statistics and Probability, Y.L. Tong Ed. IMS Lecture Notes-Monograph Series 5 (1984) 127-140.
[23] C.M. Newman, Y. Rinott and A. Tversky, Nearest neighbors and voronoi regions in certain point processes. Adv. Appl. Prob. 15 (1983) 726-751. | Zbl 0527.60050
[24] C.M. Newman and A.L. Wright, An invariance principle for certain dependent sequences. Ann. Probab. 9 (1981) 671-675. | Zbl 0465.60009
[25] V.V. Petrov, Limit theorems of probability theory: sequences of independent random variables. Clarendon Press, Oxford (1995). | MR 1353441 | Zbl 0826.60001
[26] L. Pitt, Positively Correlated Normal Variables are Associated. Ann. Probab. 10 (1982) 496-499. | Zbl 0482.62046
[27] E. Rio, Théorie asymptotique des processus aléatoires faiblement dépendants 31 (2000). | MR 2117923 | Zbl 0944.60008
[28] K.I. Sato, Lévy processes and infinitely divisible distributions. Cambridge studies in advanced mathematics 68 (1999). | MR 1739520 | Zbl 0973.60001
[29] C.M. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, in Proc. Sixth Berkeley Symp. Math. Statist. Probab. Univ. California Press 3 (1971) 583-602. | Zbl 0278.60026
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Electric Field Lines | Brilliant Math & Science Wiki
Swapnil Das, Samara Simha Reddy, and Sravanth C. contributed
Field line is a locus that is defined by a vector field and a starting location within the field. For the electric fields, we have electric field lines. As we have seen in Electrostatics, electric charges create an electric field in the space sorrounding them. It acts as a kind of "map" that gives that gives the direction and indicates the strength of the electric field at various regions in space. The concept of electric field lines was introduced by Michael Faraday, which helped him to easily visualize the electric field using intuition rather than mathematical analysis.
Definition of Electric Field Lines
Electric field lines have some important and interesting properties, let us study them.
Electric field lines always begin on a positive charge and end on a negative charge, so they do not form closed curves. They do not start or stop in midspace
The number of electric field lines leaving a positive charge or entering a negative charge is proportional to the magnitude of the charge.
Electric field lines never intersect.
In an uniform electric field, the field lines are straight, parallel and uniformly spaced.
The electric field lines can never form closed loops, as line can never start and end on the same charge.
These field lines always flow from higher potential to lower potential.
If the electric field in a given region of space is zero, electric field lines do not exist.
The tangent to a line at any point gives the direction of the electric field at the point. Also, this is the path on which a positive test charge will tend to move if free to do so.
Why don't electric field lines intersect
?
If the electric field lines intersect, then two tangents could be drawn at their point of intersection. Thus, the electric field intensity at the point will have two directions, which is absurd.
a) only a) and c) a) and b) b) only
The above diagram shows the lines of electric force and equipotential lines on a particular plane. Which of the following statements is correct?
a) The electric potential at point A is higher than that at point B.
b) The electric field strength at point A is the same as that at point B.
c) The work done by the electric force when an electrically charged particle is moved from point B to C along the equipotential line is zero.
Why aren't there any electric field lines inside a conductor
?
That is because of the fact that electric field inside a conductor is zero
!
When is an electric field said to be uniform
?
An electric field is said to be uniform, when it has the same magnitude and direction in a given region of space.
Both A and B have the same sign. If we put a positive charge at P, it will be dragged towards B. The electric field strength at point P is stronger than that at point Q. The quantity of the electric charge A is larger than that of B.
The above diagram shows electric field lines generated by two point charges A and B. Which of the following explanations is NOT correct?
Cite as: Electric Field Lines. Brilliant.org. Retrieved from https://brilliant.org/wiki/electric-field-lines/
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Monty Hall Practice Problems Online | Brilliant
You are in a game show! There are 3 closed doors: two lead to nothing and one leads to 300 dollars (there wasn't much funding for the game show that year). You get to choose a door and receive whatever is behind it. However, after you pick, the game show host (who knows where the 300 dollars are) has been instructed to open an empty door you didn't pick and sell you the chance to switch to the other unopened door (he wants to make money for next year's game show). If you want to maximize the expected value of your earnings, what is the highest price you should buy the chance to switch at?
You are in a game show! There are 10 closed doors: 9 lead to nothing and one leads to an expensive sports car. You are allowed to pick a door and earn the sports car if it's behind the door you choose. You choose a door and the host tells you he was preauthorized to make your chances of winning better! You have two options:
Option 1: Get the right to open two doors instead of one, and win if the car is behind either of the ones you open.
Option 2: Have the host open 5 empty doors (none of them the one you had chosen), and then get the right to switch if you want
You should be indifferent so long as you switch in option 2 Go with option 2, and switch Go with option 1 You should be indifferent Go with option 2, and then do not switch
A mathematician is on a gameshow, and the host gives him a choice of three doors; behind one is a Ferrari, but the other two lead to empty rooms. If he chooses the correct door, the host will open an empty door and give him the chance to choose again. However, if he chooses an incorrect door, the host will open the other empty door and give him the opportunity to choose again with probability
p
(otherwise, he will tell him that he has lost).
The mathematician picks a door and the host opens another and gives him a chance to switch. The mathematician, who always makes true statements and is aware of the host's strategy, tells the host that changing does not improve or decrease his probability of winning the Ferrari. What is
p
1
\frac{2}{3}
\frac{1}{3}
\frac{1}{2}
There are six doors lined up in a row. Two adjacent doors both have an expensive prize, and the other four have nothing. You pick the third door, but then the host (who you know will open the left-most door that is empty and you haven't chosen) opens the first door. If you then decide to switch to the fourth door, what is the probability that you got the prize?
\frac{1}{6}
\frac{1}{5}
\frac{2}{5}
\frac{1}{3}
\frac{1}{2}
You are on a game show, and must correctly identify the one door of three that has the grand prize. You pick a door, and then the host asks you to pick another door which he may or may not show you the contents of. You point to a door, and the host refuses to open it. You know that if the door you had pointed to had been the door with the prize, the host would have never opened it, and if it had been empty, he would have opened it half the time. The host then offers you the chance to switch doors (from your original choice, not the second door you pointed at) if you want, but you refuse and keep the first door you chose. What is the probability you will get the prize?
\frac{1}{3}
\frac{1}{4}
\frac{2}{3}
\frac{1}{2}
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Solve design optimization problem - MATLAB sdo.optimize - MathWorks Switzerland
\underset{p}{\text{min}}F\left(p\right)\text{ subject to }\left\{\begin{array}{l}{C}_{leq}\left(p\right)\le 0\hfill \\ {C}_{eq}\left(p\right)=0\hfill \\ A×p\le B\hfill \\ {A}_{eq}×p={B}_{eq}\hfill \\ lb\le p\le ub\hfill \end{array}
\mathit{f}\left(\mathit{x}\right)={\mathit{x}}^{2}
{\mathit{x}}^{2}-4\mathit{x}+1\le 0
\frac{2\mathit{x}}{3}-3\le 0
\mathit{f}\left(\mathit{x}\right)
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Damped free Vibration - Numerical 1 | Education Lessons
A mass of 85 kg is supported on a spring which deflects 18 mm under the weight of the mass. The vibrations of the mass are constrained to be linear and vertical. A dashpot is provided which reduces the amplitude to one-quarter of its initial value in two complete oscillations. Calculate magnitude of the damping force at unit velocity and periodic time of damped vibration.
\ m = 85
Static deflection,
\ x = 18\ \text{mm} = 0.018\ \text{m}
x_0 =
initial amplitude
x_2=
final amplitude after two complete cycle
= {1\over4}
Taking ratio,
{x_0 \over x_2}={x_0 \over {1 \over 4} x_0}=4
\text F = c.\mathring x= c
\begin{aligned} \xi= {c \over c_c} \quad \text {OR} \quad &\xi= {c \over 2m\omega_0} \quad (\text {as} \quad c_c=2m\omega_n)\\ \therefore \ &c=(2m\omega_n)\xi \ \dots\dots\dots\dots \ (1) \end{aligned}
1. The Logarithmic decrement:
\begin{aligned} \delta &= {1 \over n} log_e\bigg ({x_0 \over x_n}\bigg)\\ &= {1 \over 2} log_e\bigg ({x_0 \over x_2}\bigg)\\ &= {1 \over 2} log_e(4) \end{aligned}
2. Damping factor:
\begin{aligned} \xi &= {\delta \over \sqrt{4 \pi^2 + {\delta}^2} } \\ &={\delta \over \sqrt{4 \pi^2 + {\delta}^2} }\\ &= 0.1091 \end{aligned}
3. Frequency of undamped free vibration:
The natural circular frequency of vibration is,
\begin{aligned} \omega_n &= {\sqrt{k \over \ m}} \\ &= {\sqrt{m \ g \over \ x \ m}}\\ &= {\sqrt{g \over x}} \\ &= {\sqrt{9.81 \over 0.018}}\\ &= 23.34 \ \text {rad/s} \end{aligned}
From equation (1); we get,
Damping force at unit velocity;
\begin{aligned} \text F = c &= (2m\omega_n)\xi \\ &= (2 \ \text x \ 85\ \text x \ 23.34)\ \text x \ (0.1091) \\ &= 91.67 \ \text N \end{aligned}
The time period of damped vibration is,
\begin{aligned} t_p &= {2\pi \over \omega_d} \\ &= {2\pi \over \omega_n \ \sqrt {1 -\xi^2}} \\ &= {2\pi \over (23.34) \ \sqrt {1 -(0.1091)^2}} \\ &=0.2708 \ \text {sec} \end{aligned}
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A general effective Hamiltonian method | EMS Press
We perform a general reduction scheme that can be applied in particular to the spectral study of operators of the type
P=P(x,y,hD_x,D_y)
h
tends to zero. This scheme permits to reduce the study of
P
to the one of a semiclassical matrix operator of the type
A=A(x,hD_x)
. Here, for any fixed
(x,\xi )\in\R^n
, the eigenvalues of the principal symbol
a(x,\xi )
A
are eigenvalues of the operator
P(x,y,\xi ,D_y)
André Martinez, A general effective Hamiltonian method. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 18 (2007), no. 3, pp. 269–277
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Substitute n=1, 2, 3, 4, 5 to find the first
Substitute n=1, 2, 3, 4, 5 to find the first five sequences in the given sequence \left\{\frac{(-1)^{n-1}}{2\cdot4\cdot6\dotsm2n}\right\}
\left\{\frac{\left(-1{\right)}^{n-1}}{2\cdot 4\cdot 6\cdots 2n}\right\}
\left\{\frac{\left(-1{\right)}^{n-1}}{2\cdot 4\cdot 6\cdots 2n}\right\}
\left\{\frac{\left(-1{\right)}^{1-1}}{2\cdot 4\cdot 6\cdots 2\left(1\right)}\right\}=\frac{\left(-1{\right)}^{0}}{2}=\frac{1}{2}
\left\{\frac{\left(-1{\right)}^{2-1}}{2\cdot 4\cdot 6\cdots 2\left(2\right)}\right\}=\frac{\left(-1{\right)}^{1}}{2\cdot 4}=-\frac{1}{8}
\left\{\frac{\left(-1{\right)}^{3-1}}{2\cdot 4\cdot 6\cdots 2\left(3\right)}\right\}=\frac{\left(-1{\right)}^{2}}{2\cdot 4\cdot 6}=\frac{1}{48}
\left\{\frac{\left(-1{\right)}^{4-1}}{2\cdot 4\cdot 6\cdots 2\left(4\right)}\right\}=\frac{\left(-1{\right)}^{3}}{2\cdot 4\cdot 6\cdot 8}=-\frac{1}{384}
\left\{\frac{\left(-1{\right)}^{5-1}}{2\cdot 4\cdot 6\cdots 2\left(5\right)}\right\}=\frac{\left(-1{\right)}^{4}}{2\cdot 4\cdot 6\cdot 8\cdot 10}=\frac{1}{3840}
{a}_{n}=n{a}_{n-1}+{a}_{n-2}^{2},{a}_{0}=-1,{a}_{1}=0
\underset{n⇒\mathrm{\infty }}{lim}\frac{n}{2n-1}=\frac{1}{2}
If you flip a coin 9 times, you get a sequence of Heads (H) and tails (T).
Solve each question
(a) How many different sequences of heads and tails are possible?
(b)How many different sequences of heads and tails have exactly fiveheads?
c) How many different sequences have at most 2 heads?
{c}_{1}=4,\text{ }{c}_{2}=5,\text{ }\text{and}\text{ }{c}_{n}={c}_{n-1}.\text{ }{c}_{n-2}\text{ }\text{for}\text{ }n\ge 3
There are three sequences. Which of them are arithmetic
2,4,6,8,10,..
2,4,8,16,32,\dots
a,a+2,a+4,a+6,a+8,\dots
Which of the following sequences is NOT arithmetic?
(A) -4, 2, 8, 14, ...,
(B) 9, 4, -1, -6, ...
(C) 2, 4, 8, 16,...
\frac{1}{3},\text{ }1×\frac{1}{3},\text{ }2×\frac{1}{3},\text{ }3×\frac{1}{3}
',...
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False (logic) - Wikipedia
See also: Falsity
Find sources: "False" logic – news · newspapers · books · scholar · JSTOR (March 2012) (Learn how and when to remove this template message)
In logic, false[1] or untrue is the state of possessing negative truth value or a nullary logical connective. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation, truth.[2] Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, Opq), and the up tack symbol
{\displaystyle \bot }
Another approach is used for several formal theories (e.g., intuitionistic propositional calculus), where a propositional constant (i.e. a nullary connective),
{\displaystyle \bot }
, is introduced, the truth value of which being always false in the sense above.[5][6][7] It can be treated as an absurd proposition, and is often called absurdity.
1 In classical logic and Boolean logic
2 False, negation and contradiction
In classical logic and Boolean logicEdit
In Boolean logic, each variable denotes a truth value which can be either true (1), or false (0).
In a classical propositional calculus, each proposition will be assigned a truth value of either true or false. Some systems of classical logic include dedicated symbols for false (0 or
{\displaystyle \bot }
), while others instead rely upon formulas such as p ∧ ¬p and ¬(p → p).
In both Boolean logic and Classical logic systems, true and false are opposite with respect to negation; the negation of false gives true, and the negation of true gives false.
{\displaystyle x}
{\displaystyle \neg x}
The negation of false is equivalent to the truth not only in classical logic and Boolean logic, but also in most other logical systems, as explained below.
False, negation and contradictionEdit
In most logical systems, negation, material conditional and false are related as:
In fact, this is the definition of negation in some systems,[8] such as intuitionistic logic, and can be proven in propositional calculi where negation is a fundamental connective. Because p → p is usually a theorem or axiom, a consequence is that the negation of false (¬ ⊥) is true.
A contradiction is the situation that arises when a statement that is assumed to be true is shown to entail false (i.e., φ ⊢ ⊥). Using the equivalence above, the fact that φ is a contradiction may be derived, for example, from ⊢ ¬φ. A statement that entails false itself is sometimes called a contradiction, and contradictions and false are sometimes not distinguished, especially due to the Latin term falsum being used in English to denote either, but false is one specific proposition.
Logical systems may or may not contain the principle of explosion (ex falso quodlibet in Latin), ⊥ ⊢ φ for all φ. By that principle, contradictions and false are equivalent, since each entails the other.
Main article: Consistency
A formal theory using the "
{\displaystyle \bot }
" connective is defined to be consistent, if and only if the false is not among its theorems. In the absence of propositional constants, some substitutes (such as the ones described above) may be used instead to define consistency.
Wikiquote has quotations related to Falsehood.
^ Its noun form is falsity.
^ Jennifer Fisher, On the Philosophy of Logic, Thomson Wadsworth, 2007, ISBN 0-495-00888-5, p. 17.
^ Willard Van Orman Quine, Methods of Logic, 4th ed, Harvard University Press, 1982, ISBN 0-674-57176-2, p. 34.
^ "Truth-value | logic". Encyclopedia Britannica. Retrieved 2020-08-15.
^ George Edward Hughes and D.E. Londey, The Elements of Formal Logic, Methuen, 1965, p. 151.
^ Leon Horsten and Richard Pettigrew, Continuum Companion to Philosophical Logic, Continuum International Publishing Group, 2011, ISBN 1-4411-5423-X, p. 199.
^ Graham Priest, An Introduction to Non-Classical Logic: From If to Is, 2nd ed, Cambridge University Press, 2008, ISBN 0-521-85433-4, p. 105.
^ Dov M. Gabbay and Franz Guenthner (eds), Handbook of Philosophical Logic, Volume 6, 2nd ed, Springer, 2002, ISBN 1-4020-0583-0, p. 12.
Retrieved from "https://en.wikipedia.org/w/index.php?title=False_(logic)&oldid=1080646504"
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Spectral kurtosis from signal or spectrogram - MATLAB pkurtosis - MathWorks Nordic
Plot Spectral Kurtosis of Nonstationary Signal Using Different Confidence Levels
Plot Spectral Kurtosis Using a Customized Spectrogram
'ConfidenceLevel'p
sk = pkurtosis(x)
sk = pkurtosis(x,sampx)
sk = pkurtosis(xt)
sk = pkurtosis(___,window)
sk = pkurtosis(s,sampx,f,window)
[sk,fout] = pkurtosis(___)
[___,thresh] = pkurtosis(___,'ConfidenceLevel',p)
pkurtosis(___)
sk = pkurtosis(x) returns the spectral kurtosis of vector x as the vector sk. pkurtosis uses normalized frequency (evenly spaced frequency vector spanning [0 π]) to compute the time values. pkurtosis computes the spectrogram of x using pspectrum with default window size (time resolution in samples), and 80% window overlap.
sk = pkurtosis(x,sampx) returns the spectral kurtosis of vector x sampled at rate or time interval sampx.
sk = pkurtosis(xt) returns the spectral kurtosis of single-variable timetable xt in the vector sk. xt must contain increasing finite time samples.
sk = pkurtosis(___,window) returns the spectral kurtosis using the time resolution specified in window for the pspectrum spectrogram computation. You can use window with any of the input arguments in previous syntaxes.
sk = pkurtosis(s,sampx,f,window) returns the spectral kurtosis using the spectrogram or power spectrogram s, along with:
Sample rate or time, sampx, of the original time-series signal that was transformed to produce s
Spectrogram frequency vector f
Spectrogram time resolution window
Use this syntax when you want to customize the options for pspectrum, rather than accept the default pspectrum options that pkurtosis applies. You can specify sampx as empty to default to normalized frequency. Although window is optional for previous syntaxes, you must supply a value for window when using this syntax.
[sk,fout] = pkurtosis(___) returns the spectral kurtosis sk along with the frequency vector fout. You can use these output arguments with any of the input arguments in previous syntaxes.
[___,thresh] = pkurtosis(___,'ConfidenceLevel',p) returns the spectral kurtosis threshold thresh using the confidence level p. thresh represents the range within which the spectral kurtosis indicates a Gaussian stationary signal, at the optional confidence level p that you either specify or accept as default. Specifying p allows you to tune the sensitivity of the spectral kurtosis thresh results to behavior that is non-Gaussian or nonstationary. You can use the thresh output argument with any of the input arguments in previous syntaxes. You can also set the confidence level in previous syntaxes, but it has no effect unless you are returning or plotting thresh.
pkurtosis(___) plots the spectral kurtosis, along with confidence level and thresholds, without returning any data. You can use this syntax with any of the input arguments in previous syntaxes.
Plot the spectral kurtosis of a chirp signal in white noise, and see how the nonstationary non-Gaussian regime can be detected. Explore the effects of changing the confidence level, and of invoking normalized frequency.
Create a chirp signal, add white Gaussian noise, and plot.
x = xc + randn(1,length(t));
title('Chirp Signal with White Gaussian Noise')
Plot the spectral kurtosis of the signal.
title('Spectral Kurtosis of Chirp Signal with White Gaussian Noise')
The plot shows a clear extended excursion from 300–400 Hz. This excursion corresponds to the signal component which represents the nonstationary chirp. The area between the two horizontal red-dashed lines represents the zone of probable stationary and Gaussian behavior, as defined by the 0.95 confidence interval. Any kurtosis points falling within this zone are likely to be stationary and Gaussian. Outside of the zone, kurtosis points are flagged as nonstationary or non-Gaussian. Below 300 Hz, there are a few additional excursions slightly above the above the zone threshold. These excursions represent false positives, where the signal is stationary and Gaussian, but because of the noise, has exceeded the threshold.
Investigate the impact of the confidence level by changing it from the default 0.95 to 0.85.
pkurtosis(x,fs,'ConfidenceLevel',0.85)
title('Spectral Kurtosis of Chirp Signal with Noise at Confidence Level of 0.85')
The lower confidence level implies more sensitive detection of nonstationary or non-Gaussian frequency components. Reducing the confidence level shrinks the thresh-delimited zone. Now the low-level excursions — false alarms — have increased in both number and amount. Setting the confidence level is a balancing act between achieving effective detection and limiting the number of false positives.
You can accurately determine and compare the zone width for the two cases by using the pkurtosis form that returns it.
[sk1,~,thresh95] = pkurtosis(x);
[sk2,~,thresh85] = pkurtosis(x,'ConfidenceLevel',0.85);
thresh = [thresh95 thresh85]
Plot the spectral kurtosis again, but this time, omit the sample time information so that pkurtosis plots normalized frequency.
pkurtosis(x,'ConfidenceLevel',0.85)
title('Spectral Kurtosis using Normalized Frequency')
The frequency axis has changed from Hz to a scale from 0 to π rad/sample.
When using signal input data, pkurtosis generates a spectrogram by using pspectrum with default options. You can also create the spectrogram yourself if you want to customize the options.
Generate a spectrogram that uses your specification for window, overlap, and number of FFT points. Then use that spectrogram in pkurtosis.
overlap = round(window*0.8);
nfft = 2*window;
[s,f,t] = spectrogram(x,window,overlap,nfft,fs);
pkurtosis(s,fs,f,window)
The magnitude of the excursion is higher, and therefore better differentiated, than with default inputs in previous examples. However, the excursion magnitude here is not as high as it is in the kurtogram-optimized window example.
Time-series signal from which pkurtosis returns the spectral kurtosis, specified as a vector.
Sample rate or sample time, specified as one of the following::
\frac{1}{100}<\frac{\text{Median time interval}}{\text{Mean time interval}}<100.
If you specify sampx as empty, then pkurtosis uses normalized frequency. In other words, it assumes an evenly spaced frequency vector spanning [0 π].
Signal timetable from which pkurtosis returns the spectral kurtosis, specified as a timetable that contains a single variable with a single column. xt must contain increasing, finite row times. If the timetable has missing or duplicate time points, you can fix it using the tips in Clean Timetable with Missing, Duplicate, or Nonuniform Times. xt can be nonuniformly sampled, with the pspectrum constraint that the median time interval and the mean time interval must obey:
\frac{1}{100}<\frac{\text{Median time interval}}{\text{Mean time interval}}<100.
window — Window time resolution
Window time resolution to use for the internal pspectrum spectrogram computation, specified as a positive scalar in samples. window is required for syntaxes that use an existing spectrogram as input, and optional for the rest. You can use the function kurtogram to determine the optimal window size to use. pspectrum uses 80% overlap by default.
s — Spectrogram or power spectrogram of signal
complex matrix | real nonnegative matrix
Power spectrogram or spectrum of a signal, specified as a matrix (spectrogram) or a column vector (spectrum).
If s is complex, then pkurtosis treats s as a short-time Fourier transform (STFT) of the original signal (spectrogram).
If s is real, then pkurtosis treats s as the square of the absolute values of the STFT of the original signal (power spectrogram). Thus, every element of s must be nonnegative.
If you specify s, pkurtosis uses s rather than generate its own spectrogram or power spectrogram. For an example, see Plot Spectral Kurtosis Using a Customized Spectrogram.
f — Frequencies for s
Frequencies for spectrogram or power spectrogram s when s is supplied explicitly to pkurtosis, specified as a vector in hertz. The length of f must be equal to the number of rows in s.
'ConfidenceLevel', p — Confidence level
0.95 (default) | [0 to 1]
Confidence level used to determine whether signal is likely to be Gaussian and stationary, specified as a numeric scalar value from 0 to 1. p influences the thresh range where the spectral kurtosis value indicates a Gaussian and stationary signal. The confidence level therefore provides a detection-sensitivity tuning parameter. Kurtosis values outside of this range indicate, with a probability of (1-p), non-Gaussian or nonstationary behavior. For an example, see Plot Spectral Kurtosis of Nonstationary Signal Using Different Confidence Levels.
sk — Spectral kurtosis
Spectral Kurtosis, returned as a double vector. The spectral kurtosis is a statistical quantity that contains low values where data is stationary and Gaussian, and high values where transients occur. One use of the spectral kurtosis is to detect and locate nonstationary or non-Gaussian behavior that could result from faults or degradation. The high-valued kurtosis data reveals such signal components.
fout — frequencies for sk
Frequencies associated with sk values, returned as a vector in hertz.
thresh — Spectral kurtosis band size for stationary Gaussian behavior
Spectral kurtosis band size for stationary Gaussian behavior, specified as a numeric scalar representing the thickness of the band centered at the sk = 0 line, given confidence level p. Excursions outside the thresh-delimited band indicate possible nonstationary or non-Gaussian behavior. Confidence level p directly influences the thickness of the band and the sensitivity of the results. For an example, see Plot Spectral Kurtosis of Nonstationary Signal Using Different Confidence Levels.
Spectral kurtosis (SK) is a statistical tool that can indicate and pinpoint nonstationary or non-Gaussian behavior in the frequency domain, by taking:
Small values at frequencies where stationary Gaussian noise only is present
High positive values at frequencies where transients occur
This capability makes SK a powerful tool for detecting and extracting signals associated with faults in rotating mechanical systems. On its own, SK can identify features or conditional indicators for fault detection and classification. As preprocessing for other tools such as envelope analysis, SK can supply key inputs such as optimal band [2], [1].
The spectral kurtosis, or K(f), of a signal x(t) can be computed based on the short-time Fourier transform (STFT) of the signal, S(t,f):
S\left(t,f\right)=\underset{-\infty }{\overset{+\infty }{\int }}x\left(t\right)w\left(t-\tau \right){e}^{-2\pi ft}dt,
where w(t) is the window function used in STFT. K(f) is calculated as:
K\left(f\right)=\frac{〈{|S\left(t,f\right)|}^{4}〉}{{〈{|S\left(t,f\right)|}^{2}〉}^{2}}-2,f\ne 0,
〈·〉
is the time-average operator.
If the signal x(t) contains only stationary Gaussian noise, then K(f) at each frequency f has an asymptotic normal distribution with 0 mean and variance 4/M , where M is the number of elements along the time axis in S(t,f). Hence, a statistical threshold
{s}_{\alpha }
given a confidence level α is:
{s}_{\alpha }={\Phi }^{-1}\left(\alpha \right)\frac{2}{\sqrt{M}},
{\Phi }^{-1}
is the quantile function of the standard normal distribution.
It is important to note that the STFT window length Nw directly drives frequency resolution, which is fs/Nw, where fs is the sample rate. The window size must be shorter than the spacing between transient impulses, but longer than the individual transient impulses.
[1] Antoni, J. "The Spectral Kurtosis: A Useful Tool for Characterising Non-Stationary Signals." Mechanical Systems and Signal Processing. Vol. 20, Issue 2, 2006, pp. 282–307.
[2] Antoni, J., and R. B. Randall. "The Spectral Kurtosis: Application to the Vibratory Surveillance and Diagnostics of Rotating Machines." Mechanical Systems and Signal Processing. Vol. 20, Issue 2, 2006, pp. 308–331.
If the input is a pure tone signal, the result may show small numerical differences with respect to in-memory computation of spectral kurtosis.
kurtogram | pentropy | pspectrum
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Box Office Math | Media4Math
In this module, students explore algebraic expressions to model different quantities. They look at expressions that involve addition, subtraction, and multiplication. Then they look at real world data from the Star Wars movies since the Disney acquisition of the franchise. Students analyze whether the purchase of the Star Wars franchise has been profitable for Disney.
For this lesson make sure that students are familiar with the definitions of variables, unknowns, and constants. Review definitions are provided.
Students will learn about modeling and evaluating algebraic expressions. In particular students will look at an expression of the form
px-C
, where p is the price and C is the cost of putting on an event (concert, movie).
Students then look at a real-world application of the Disney purchase of the Star Wars franchise. Students analyze box office data and arrive at an algebraic expression using this data set.
This lesson addresses Common Core standards from grades 6 and 7. This lesson can be assigned to individual students or teams of students. The lesson can be completed in about 20 to 25 minutes.
How to model and evaluate algebraic expressions
Using algebraic expression to mode real word situations
Analyze real world data
Knowledge of variables, constants, and unknowns
CCSS.MATH.CONTENT.6.EE.A.2, CCSS.MATH.CONTENT.7.EE.B.3
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What is the domain of the function g\left(x\right)=\frac{\mathrm{cos}\left[x-2\right]}{\left[x-1\right]\left[x-2\right]}?\phantom{\rule{0ex}{0ex}}\left(A\right) \left(-\infty ,1\right)\cup \left[3,\infty \right)\phantom{\rule{0ex}{0ex}}\left(B\right) \left(-\infty ,1\right)\cup \left(3,\infty \right)\phantom{\rule{0ex}{0ex}}\left(C\right) \left[0,1\right)\phantom{\rule{0ex}{0ex}}\left(D\right) \left[1,3\right)
The domain of the function :
\mathrm{f}\left(\mathrm{x}\right)=\sqrt{1-{\mathrm{log}}_{\mathrm{e}}\left(1-2\mathrm{x}\right)} \mathrm{is}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{A}\right) \left(-\infty ,\frac{1-\mathrm{e}}{2}\right)\cup \left(\frac{1}{2},\infty \right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{B}\right) \left[\frac{1-\mathrm{e}}{2},\frac{1}{2}\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{C}\right) \left(\frac{1-\mathrm{e}}{2},\frac{1}{2}\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{D}\right) \left(\frac{1-\mathrm{e}}{2},\frac{1}{2}\right]
Q. The domain of the function
f\left(x\right)=\sqrt{\frac{2-\left|x\right|}{3-\left|x\right|}}
\left(\mathrm{A}\right) \left(-3,3\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{B}\right) \left(-2,2\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{C}\right) \left(-\infty ,-3\right]\cup \left[-2,2\right]\cup \left[3,\infty \right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{D}\right) \left(-\infty ,-3\right)\cup \left[-2,2\right]\cup \left(3,\infty \right)\phantom{\rule{0ex}{0ex}}
Q. Let f be a real valued function defined by
\mathrm{f}\left(\mathrm{x}\right)=\frac{\left[\left(\mathrm{x}-1\right)\left(\mathrm{x}-2\right)\right]\mathrm{In}\left(\left[\left(\mathrm{x}-1\right)\left(\mathrm{x}-2\right)\right]\right)}{\sqrt{-\left(\mathrm{x}+3\right)\left(\mathrm{x}-4\right)}}
Where[.] denotes the greatest integer function, then domain of the function is.
\left(\mathrm{A}\right) \left(-2,\frac{3-\sqrt{5}}{2}\right)\cup \left(\frac{3+\sqrt{5}}{2},4\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{B}\right) \left(-3,\frac{3-\sqrt{5}}{2}\right] \cup \left[\frac{3+\sqrt{5}}{2},3\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{C}\right) \left(-3,\frac{3-\sqrt{5}}{2}\right] \cup \left[\frac{3+\sqrt{5}}{2},4\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{D}\right) \left(-3,\frac{3+\sqrt{5}}{2}\right]\cup \left[\frac{3+\sqrt{5}}{2},4\right)\phantom{\rule{0ex}{0ex}}
Q. The number of solution of the equation [x]+[-x]+x2=1, where [.] denotes the greatest integer function is.
(A) Infinitely many
If A = {x: x is a real number} and B = {0}, then the set (B
×
A) is
A. the set containing the origin in two-dimensional space
B. the set of all points lying on the x-axis in two-dimensional space
C. the set of all points lying on the y-axis in two-dimensional space
D. the set of all points on a coordinate plane in two-dimensional space
Q. If sec A + tan A = 13/3, then sec A is equal to
The tips of the pendulums of two wall clocks having their respective lengths as 35 cm and 49 cm make arcs of lengths 11 cm and 22 cm, respectively.
What are the respective angles through which the pendulums of the two wall clocks will swing?
\left(\mathrm{A}\right) \frac{\mathrm{\pi }}{4}\mathrm{radian} \mathrm{and} \frac{\mathrm{\pi }}{3}\mathrm{radian}\phantom{\rule{0ex}{0ex}}\left(\mathrm{B}\right) \frac{\mathrm{\pi }}{10}\mathrm{radian} \mathrm{and} \frac{\mathrm{\pi }}{7}\mathrm{radian}\phantom{\rule{0ex}{0ex}}\left(\mathrm{C}\right) 4\mathrm{\pi } \mathrm{radian} \mathrm{and} 3\mathrm{\pi } \mathrm{radian}\phantom{\rule{0ex}{0ex}}\left(\mathrm{D}\right) \frac{\mathrm{\pi }}{3}\mathrm{radian} \mathrm{and} \frac{\mathrm{\pi }}{4}\mathrm{radian}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}
\mathrm{Q}. \mathrm{Find} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{expression} \sqrt{2 \mathrm{tan} \mathrm{x}+\frac{1}{{\mathrm{cos}}^{2}\mathrm{x}}}, \forall \frac{7\mathrm{\pi }}{2}<\mathrm{x}<\frac{15\mathrm{\pi }}{4}\phantom{\rule{0ex}{0ex}}\left(\mathrm{a}\right) -\left(1+\mathrm{tan} \mathrm{x}\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{b}\right) \left(1-\mathrm{tan} \mathrm{x}\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{c}\right) \left(1+\mathrm{tan} \mathrm{x}\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{d}\right) \left(\mathrm{tan} \mathrm{x}-1\right)
The number of solutions of the equation
\left[x\right]+\left[-x\right]+{x}^{2}=1
, where [.] denotes the greatest integer function is
A. Infinitely many
\mathrm{f}\left(\mathrm{x}\right) =\frac{\left[\left(\mathrm{x}-1\right)\left(\mathrm{x}-2\right)\right]\mathrm{in}\left(\left[\left(\mathrm{x}-1\right)\left(\mathrm{x}-2\right)\right]\right) }{\sqrt{-\left(\mathrm{x}+3\right)\left(\mathrm{x}-4\right)}} \mathrm{where} \left[.\right] \mathrm{denotes} \mathrm{the} \mathrm{greatest} \mathrm{integer} \mathrm{function}, \mathrm{then} \mathrm{domain} \mathrm{of} \mathrm{the} \mathrm{function} \mathrm{is}. \phantom{\rule{0ex}{0ex}}\left(\mathrm{A}\right) \left(-2,\frac{3-\sqrt{5}}{2}\right)\cup \left(\frac{3+\sqrt{5}}{2},4\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{B}\right) \left(-3,\frac{3-\sqrt{5}}{2}\right)\cup \left(\frac{3+\sqrt{5}}{2},3\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{C}\right) \left(-3,\frac{3-\sqrt{5}}{2}\right)\cup \left(\frac{3+\sqrt{5}}{2},4\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{D}\right) \left(-3,\frac{3+\sqrt{5}}{2}\right)\cup \left(\frac{3+\sqrt{5}}{2},4\right)
Q). The domain of the function
f\left(x\right)=\sqrt{\frac{2-\left|x\right|}{3-\left|x\right|}}
B. (- 2, 2)
\left(-\infty ,-3\right)\cup \left[2, 2\right]\cup \left[3, \infty \right)
\left(-\infty ,-3\right)\cup \left[2, 2\right]\cup \left(3, \infty \right)
\mathrm{Question} \mathrm{No}. 7\phantom{\rule{0ex}{0ex}}\mathrm{The} \mathrm{range} \mathrm{of} \mathrm{the} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right) =\sqrt{5\mathrm{x}-{\mathrm{x}}^{2}-6} \mathrm{is}. \phantom{\rule{0ex}{0ex}}\left(\mathrm{A}\right) \left[-\frac{1}{2},\frac{1}{2}\right]\phantom{\rule{0ex}{0ex}}\left(\mathrm{B}\right) \left[0,\frac{1}{2}\right]\phantom{\rule{0ex}{0ex}}\left(\mathrm{C}\right) \left(0,\frac{1}{2}\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{D}\right) \left(\frac{1}{2},-\frac{1}{2}\right)
Question NO.6 ) If {( – 1, 16), (0, 1), (1, 4) , (2, 25)}
\subset
f, where f : R
\to
R, is a quadratic function, then the function f(x) is
(A) 9x2 + 6x +1
\mathrm{f}\left(\mathrm{x}\right)=\sqrt{1-{\mathrm{log}}_{\mathrm{e}}\left(1-2\mathrm{x}\right)} \mathrm{is}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\left(\mathrm{A}\right) \left(-\infty , \frac{1-\mathrm{e}}{2}\right)\cup \left(\frac{1}{2},\infty \right)\phantom{\rule{0ex}{0ex}}\left(\mathbf{B}\right) \left[\frac{1-\mathrm{e}}{2}, \frac{1}{2}\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{C}\right) \left(\frac{1-\mathrm{e}}{2}, \frac{1}{2}\right)\phantom{\rule{0ex}{0ex}}\left(\mathrm{D}\right) \left(\frac{1-\mathrm{e}}{2}, \frac{1}{2}\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}
|
Absolute value and complex magnitude - MATLAB abs - MathWorks Deutschland
Absolute Value of Scalar
Absolute Value of Vector
Y = abs(X) returns the absolute value of each element in array X.
If X is complex, abs(X) returns the complex magnitude.
y = abs(-5)
Create a numeric vector of real values.
x = [1.3 -3.56 8.23 -5 -0.01]'
Find the absolute value of the elements of the vector.
y = abs(3+4i)
Input array, specified as a scalar, vector, matrix, or multidimensional array. If X is complex, then it must be a single or double array. The size and data type of the output array is the same as the input array.
The absolute value (or modulus) of a real number is the corresponding nonnegative value that disregards the sign.
For a real value, a, the absolute value is:
a, if a is greater than or equal to zero
-a, if a is less than zero
The complex magnitude (or modulus) is the length of a vector from the origin to a complex value plotted in the complex plane.
For a complex value,
|a+bi|
\sqrt{{a}^{2}+{b}^{2}}
angle | sign | unwrap | hypot | norm | imag | real
|
Conductors Practice Problems Online | Brilliant
If the total charge on the surface of a conducting sphere is
2.3 \,\mu\text{C}
and the diameter of the sphere is
1.1\text{ m},
what is the magnitude of the electric field just outside the surface of the sphere, due to the surface charge?
\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.
2.8 \times 10^4 \text{ N/C}
6.9 \times 10^4 \text{ N/C}
3.4 \times 10^4 \text{ N/C}
4.6 \times 10^4 \text{ N/C}
Suppose that a charged particle is held at the center of two concentric conducting spherical shells
A
B,
as shown in the above figure. The radius of sphere
A
r_A=2\text{ cm}
and that of sphere
B
r_B=4\text{ cm}.
The net flux
\Phi
through a Gaussian sphere centered on the particle is
\begin{aligned} -16.0 \times 10^5 \text{ N}\cdot\text{m}^2\text{/C} &\text{for } 0 < r < r_A \\ +8.0 \times 10^5 \text{ N}\cdot\text{m}^2\text{/C} &\text{for } r_A < r < r_B\\ -4.0 \times 10^5 \text{ N}\cdot\text{m}^2\text{/C} &\text{for } r > r_A, \end{aligned}
r
is the radius of the Gaussian sphere. Then what are the net charges of shell
A
and shell
B,
\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.
q_A=-21.3\,\mu\text{C},q_B=3.5\,\mu\text{C}
q_A=21.3\,\mu\text{C},q_B=-14.2\,\mu\text{C}
q_A=14.2\,\mu\text{C},q_B=-10.6\,\mu\text{C}
q_A=21.3\,\mu\text{C},q_B=-10.6\,\mu\text{C}
If an isolated conductor has net charge
+15.0 \times 10^{-6}\text{ C}
and a cavity with a point charge
q=+4.0 \times 10^{-6}\text{ C},
what are the charge of the cavity wall
q_w
and the charge on the outer surface
q_s?
q_w=-4.0\,\mu\text{C}, q_s=19.0\,\mu\text{C}
q_w=4.0\,\mu\text{C}, q_s=15.0\,\mu\text{C}
q_w=4.0\,\mu\text{C}, q_s=11.0\,\mu\text{C}
q_w=-4.0\,\mu\text{C}, q_s=11.0\,\mu\text{C}
Consider a uniformly charged conducting sphere. If the radius of the sphere is
0.50\text{ m}
8.40 \,\mu\text{C/m}^2,
what is the approximate total electric flux leaving the surface of the sphere?
\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.
\num{2.98e6} \text{ N}\cdot\text{m}^2\text{/C}
\num{1.49e6} \text{ N}\cdot\text{m}^2\text{/C}
\num{3.58e6} \text{ N}\cdot\text{m}^2\text{/C}
\num{4.47e6} \text{ N}\cdot\text{m}^2\text{/C}
If the magnitude of electric field just above the surface of a charged conducting cylinder is
2.8 \times 10^5 \text{ N/C},
what is the surface charge density of the cylinder?
\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.
7.5 \times 10^{-7} \text{ C/m}^2
5.0 \times 10^{-6} \text{ C/m}^2
2.5 \times 10^{-6} \text{ C/m}^2
10.0 \times 10^{-7} \text{ C/m}^2
|
LMIs in Control/pages/H2 index - Wikibooks, open books for an open world
LMIs in Control/pages/H2 index
H2 Index Deduced LMI
Although there are ways to evaluate an upper bound on the H2, the verification of the bound on the H2-gain of the system can be done via the deduced condition.
4 The LMI - Deduced Conditions for H2-norm
We consider the generalized Continuous-Time LTI system with the state space realization of
{\displaystyle (A,B,C,D)}
{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)\end{aligned}}}
{\displaystyle x(t)\in \mathbb {R} ^{n}}
{\displaystyle y(t)\in \mathbb {R} ^{m}}
{\displaystyle u(t)\in \mathbb {R} ^{r}}
are the system state, output, and the input vectors respectively.
{\displaystyle {\begin{aligned}G(s)=C(sI-A)^{-1}B+D\end{aligned}}}
{\displaystyle A,B,C}
{\displaystyle \gamma >0}
(a given scalar), the transfer function satisfies
{\displaystyle \left\|C(sI-A)^{-1}B+D\right\|_{2}<\gamma }
The H2-norm condition on Transfer function holds only when the matrix A is stable. And this can be conveniently converted to an LMI problem
if and only if 1. There exists a symmetric matrix
{\displaystyle X>0}
{\displaystyle AX+XA^{T}+BB^{T}<0}
{\displaystyle trace(CXC^{T})<\gamma ^{2}}
2. There exists a symmetric matrix
{\displaystyle Y>0}
{\displaystyle AY+YA^{T}+C^{T}C<0}
{\displaystyle trace(B^{T}YB)<\gamma ^{2}}
The LMI - Deduced Conditions for H2-norm Edit
These deduced condition can be derived from the above equations. According to this
{\displaystyle \gamma >0}
{\displaystyle \left\|G(s)\right\|_{2}<\gamma }
{\displaystyle Z}
{\displaystyle P}
; and a matrix
{\displaystyle V}
{\displaystyle trace(Z)<\gamma ^{2}}
{\displaystyle {\begin{bmatrix}-Z&B^{T}\\B&-P\end{bmatrix}}}
{\displaystyle {\begin{aligned}<0\end{aligned}}}
{\displaystyle {\begin{bmatrix}-(V+V^{T})&V^{T}A^{T}+P&V^{T}C^{T}&V^{T}\\AV+P&-P&0&0\\CV&0&-I&0\\V&0&0&-P\end{bmatrix}}}
{\displaystyle {\begin{aligned}<0\end{aligned}}}
{\displaystyle \gamma }
to find the minimum upper bound on the H2 gain of
{\displaystyle G(s)}
{\displaystyle \gamma }
upper bounds the norm of the system G(s).
To solve the feasibility LMI, YALMIP toolbox is required for setting up the problem, and SeDuMi or MOSEK is required to solve the problem. The following link showcases an example of the problem:
Deduced LMIs for H-infinity index
Retrieved from "https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/pages/H2_index&oldid=4052229"
|
\mathrm{restart}
\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right):
c
f
f\left(x\right)=\underset{n=0}{\overset{\infty }{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∑}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{a}_{n}{\left(x-c\right)}^{n}
{a}_{n}
{a}_{n}=\frac{{f}^{\left(n\right)}\left(c\right)}{n!}
{f}^{\left(n\right)}\left(c\right)
is the nth derivative of
c
f
c
{ⅇ}^{x}
{ⅇ}^{x}=\underset{n=0}{\overset{\infty }{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∑}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{x}^{n}}{n!}
ⅇ=\underset{n=0}{\overset{\infty }{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{∑}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{1}{n!}
\mathrm{TaylorApproximation}\left({x}^{7}-5{x}^{5}+4{x}^{4}-7{x}^{2}+3,x=1,\mathrm{order}=3\right)
\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{11}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{15}\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}{x}}^{\textcolor[rgb]{0,0,1}{3}}
x=1
{x}^{7}-5{x}^{5}+4{x}^{4}-7{x}^{2}+3
{x}^{3}-15{x}^{2}+11x-1
\mathrm{TaylorApproximation}\left({x}^{7}-5{x}^{5}+4{x}^{4}-7{x}^{2}+3,x=1,\mathrm{order}=3,\mathrm{output}=\mathrm{plot}\right)
\mathrm{TaylorApproximation}\left(\mathrm{sin}\left(x\right),x=1,\mathrm{output}=\mathrm{animation},\mathrm{order}=1..16\right)
I
-I
1
\mathrm{TaylorApproximation}\left(\mathrm{arctan}\left(x\right),x=0,\mathrm{output}=\mathrm{animation},\mathrm{order}=1..20\right)
\mathrm{TaylorApproximationTutor}\left(\right)
\mathrm{FunctionChart}\left({x}^{4}+2{x}^{3}-9{x}^{2}-3x+6,x=-5..4\right)
\mathrm{FunctionChart}\left(\mathrm{sin}\left(x\right),x=0..2\mathrm{π}\right)
\mathrm{FunctionChart}\left(\frac{{x}^{3}-2{x}^{2}-4x+2}{x-4},x=-3..3\right)
\mathrm{CurveAnalysisTutor}\left(\right)
and an expression
f\left(x\right)
x
a
f\left(a\right)
f\left(x\right)
\mathrm{Tangent}\left(f\left(x\right),x=a,\mathrm{output}=\mathrm{line}\right)
\left(\frac{\textcolor[rgb]{0,0,1}{ⅆ}}{\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{a}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\left(\textcolor[rgb]{0,0,1}{a}\right)\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{f}\left(\textcolor[rgb]{0,0,1}{a}\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}{a}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\left(\textcolor[rgb]{0,0,1}{a}\right)\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}
\mathrm{collect}\left(\mathrm{convert}\left(,\mathrm{D}\right),\mathrm{D}\left(f\right)\left(a\right)\right)
\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{a}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{D}}\left(\textcolor[rgb]{0,0,1}{f}\right)\left(\textcolor[rgb]{0,0,1}{a}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{f}\left(\textcolor[rgb]{0,0,1}{a}\right)
\mathrm{solve}\left(=0,x\right)
\frac{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{f}\left(\textcolor[rgb]{0,0,1}{a}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{D}}\left(\textcolor[rgb]{0,0,1}{f}\right)\left(\textcolor[rgb]{0,0,1}{a}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{a}}{\textcolor[rgb]{0,0,1}{\mathrm{D}}\left(\textcolor[rgb]{0,0,1}{f}\right)\left(\textcolor[rgb]{0,0,1}{a}\right)}
\mathrm{expand}\left(\right)
\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{f}\left(\textcolor[rgb]{0,0,1}{a}\right)}{\textcolor[rgb]{0,0,1}{\mathrm{D}}\left(\textcolor[rgb]{0,0,1}{f}\right)\left(\textcolor[rgb]{0,0,1}{a}\right)}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{a}
F\left(x\right)={x}^{2}-1
x=2.0
F:=x→{x}^{2}-1
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{x}\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}{1}
\mathrm{aroot}:=2.0-\frac{F\left(2.0\right)}{\mathrm{D}\left(F\right)\left(2.0\right)}
\textcolor[rgb]{0,0,1}{\mathrm{aroot}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{1.250000000}
9
\mathbf{for}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}i\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{to}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}5\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{do}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{aroot}:=\mathrm{aroot}-\frac{F\left(\mathrm{aroot}\right)}{\mathrm{D}\left(F\right)\left(\mathrm{aroot}\right)}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end do}
\textcolor[rgb]{0,0,1}{\mathrm{aroot}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{1.025000000}
\textcolor[rgb]{0,0,1}{\mathrm{aroot}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{1.000304878}
\textcolor[rgb]{0,0,1}{\mathrm{aroot}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{1.000000046}
\textcolor[rgb]{0,0,1}{\mathrm{aroot}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{1.000000000}
\textcolor[rgb]{0,0,1}{\mathrm{aroot}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{1.000000000}
\mathrm{NewtonsMethod}\left(F\left(x\right),x=2,\mathrm{output}=\mathrm{sequence}\right)
\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.250000000}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.025000000}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.000304878}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.000000046}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.000000000}
\mathrm{NewtonsMethod}\left(F\left(x\right),x=2,\mathrm{output}=\mathrm{plot}\right)
\mathrm{NewtonsMethod}\left(\frac{\mathrm{sin}\left(x\right)}{x},x=1,\mathrm{output}=\mathrm{plot}\right)
\mathrm{NewtonsMethod}\left(\frac{\mathrm{sin}\left(x\right)}{x},x=2,\mathrm{output}=\mathrm{plot}\right)
30
7
\mathrm{Digits}:=30
\mathrm{NewtonsMethod}\left({x}^{4}-4{x}^{3}+4{x}^{2}-3x+3,x=1,\mathrm{output}=\mathrm{sequence},\mathrm{iterations}=10\right)
\mathrm{Digits}:=10
\textcolor[rgb]{0,0,1}{\mathrm{Digits}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{30}
\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.33333333333333333333333333333}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28318584070796460176991150443}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231623816647766759714731909}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231595363411166690275078928}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231595363408116582940754743}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231595363408116582940754709}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231595363408116582940754707}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231595363408116582940754707}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231595363408116582940754707}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.28231595363408116582940754707}
\textcolor[rgb]{0,0,1}{\mathrm{Digits}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{10}
\mathrm{NewtonsMethodTutor}\left(\right)
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Spanning Trees Practice Problems Online | Brilliant
What are the 3rd and 5th edges that Kruskal's algorithm includes for the graph above?
E-F and F-H E-F and A-H C-H and A-H F-H and C-G
If we knew that all our input graphs have edges in the range
1
K
K
is a constant). If we use the most appropriate sorting algorithm, what is the most descriptive complexity of the overall run-time?
O(K\log(V))
\O(V\log(V)))
O(K^2 + V^2 + E^2)
O(E+K+V\log(V))
How many spanning trees does the following graph have?
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A Modified Approach to the New Solutions of Generalized mKdV Equation Using -Expansion
Yu Zhou, Ying Wang, "A Modified Approach to the New Solutions of Generalized mKdV Equation Using -Expansion", International Scholarly Research Notices, vol. 2014, Article ID 856396, 10 pages, 2014. https://doi.org/10.1155/2014/856396
Yu Zhou 1 and Ying Wang1,2
Academic Editor: G. Mishuris
The modified -expansion method is applied for finding new solutions of the generalized mKdV equation. By taking an appropriate transformation, the generalized mKdV equation is solved in different cases and hyperbolic, trigonometric, and rational function solutions are obtained.
The evolutions of the physical, engineering, and other systems always behave nonlinearly; hence many nonlinear evolution equations have been introduced to interpret the phenomena. Many kinds of mathematical methods have been established to investigate the solutions of those nonlinear evolution equations both numerically and asymptotically, while the exact solutions are of particular interests. In recent decades, with the rapid progress of computation methods, many effective calculating approaches have been developed, for example, the tanh-coth expansion [1, 2], -expansion [3, 4], Painlevé expansion [5], Jacobi elliptic function method [6], Hirota bilinear transformation [7], Backlund/Darboux transformation [8, 9], variational method [10], the homogeneous balance method [11], exp-function expansion [12], and so on. However, a unified approach to obtain the complete solutions of the nonlinear evolution equations has not been revealed.
Within recent years, a new method called ()-expansion [13] has been proposed for finding the traveling wave solutions of the nonlinear evolution equations. Many equations have been investigated and many solutions have been found using the method, including KdV equation, Hirota-Satsuma equation [13], coupled Boussinesq equation [14], generalized Bretherton equation [15], the mKdV equation [16], the Burgers-KdV equation, the Benjamin-Bona-Mahony equation [17], the Whitham-Broer-Kaup-like equation [18], the Kolmogorov-Petrovskii-Piskunov equation [19], KdV-Burgers equation [20], and Drinfeld-Sokolov-Satsuma-Hirota equation [21].
The mKdV equation, a modified version of the Korteweg-de Vries (KdV) equation, has been investigated extensively since Zabusky showed how this equation depict the oscillations of a lattice of particles connected by nonlinear springs as the Fermi-Pasta-Ulam (FPU) model [22–25]. Afterwards, this equation has been used to describe the evolution of internal waves at the interface of two layers of equal depth [26]. Generally, the KdV theory describes the weak nonlinearity and weak dispersion while, in the study of nonlinear optics, the complex mKdV equation has even been used to describe the propagation of optical pulses in nematic optical fibers when we go beyond the usual weakly nonlinear limit of Kerr medium [27]. In some cases, the exponential order may be not a positive integer, but just a real number. After this kind of generalized mKdV equation [28] has been introduced, interests of investigating the solutions of it [29] have been inspired; then the standard expansion methods cannot be applied, and some kinds of transformation are needed.
In this paper, we modify the standard ()-expansion method and use it to solve the generalized mKdV equation. In next section, we briefly introduce the modified ()-expansion method while in Section 3, we apply it to find some types of new solutions of mKdV equation, and the last section gives the summary and conclusion.
2. An Introduction to the Modified ()-Expansion Method
Recently, a new approach called ()-expansion has been proposed dealing with the problems of finding solutions of nonlinear evolution equations [13] and some modifications to this method have been developed. Here we briefly outline the main steps of the modified ()-expansion method in the following.
Step 1. We consider a given where is a polynomial for its arguments and is the unknown function. Introducing the new variable and supposing that and , hence, the partial differential equation (PDE) (1) is reduced to an ordinary differential equation (ODE) for as
Step 2. For ODE (2) above, the solution could be expressed by a polynomial in as where is the solution of a second order linear ODE with constants , to be determined later. Positive integer is an index yet undetermined which should be calculated by the balance between the highest order derivatives and the nonlinear terms from ODE (2). By solving (4), it is apparent that the form of in three different cases read as follows.
When , where and in above three solutions (5), (6), and (7) of (4) are integrate constants.
Step 3. Substituting solution (3) into ODE (2) using (4), we have a set of differential equations. Collecting all terms together according to the same order of , the left-hand side of ODE (2) becomes a long expression in the form of polynomial of . Making all coefficients of each order of equal to zero, the solution sets of the parameters , , , and will be got after solving the algebra equations.
Step 4. Based on the last step, we now have the solutions of the coefficient algebra equations and after substituting the parameters , , and so forth into solution (3), we could reach different types of travelling wave solutions of the PDE (1).
3. Application to the Generalized mKdV Equation
Now, we consider the generalized mKdV equation with the form [28] while parameters and are real constants. Denoting the travelling wave solution as with , then the PDE (8) becomes an ODE. After integrating with single variable and setting the integrate constant to zero, we have It can be easily seen that the standard -expansion cannot be applied directly to this situation because of the arbitrary power index , which results in the noninteger power index of . So it is necessary to introduce a transformation to deal with it via assuming that then (10) becomes The balancing between the highest order nonlinear term and the highest order derivative term leads to the balance parameter ; hence the solution (3) could be expressed as with and , , , , and being constants to be determined later.
Substituting (13) into (12), making use of (4), a polynomial of is obtained; a set of nonlinear algebra equations about undetermined constants , , , , , , , and are reached through setting the coefficients of each order of to zero. These equations are expressed as follows.
-order: Equations for the coefficients of () are similar to the above equations and hence not shown here.
It is straight forward to give the solution sets of the algebraic equations in different cases as follows.
Case i. All the coefficients in (13) are equal to 0, and , , and are arbitrary.
Case ii. All the coefficients in (13) are equal to 0 except for , and , , are arbitrary.
Case iii. All the coefficients in (13) equal to 0 except for .
Cases i to iii are trivial and of no interest, hence not discussed here. We focus our attention to cases from iv to x. Using solutions (21) to (27), solution (13) can be expressed in different forms corresponding to different cases listed above.
For Case iv, there are two solution types with .
(iv-1) When , we obtain the hyperbolic function solution: where and are integration constants. Recalling (11) we get For simplicity, we only show expression for rather than in the following cases.
(iv-2) When , we have the trigonometric function solution:
For Case v, there are also two types of solution with .
(v-1) When :
For Case vi, there are three types of solution with .
(vi-1) When :
For Case vii, we only have one type of solution: where .
For Case viii, there are three types of solution with .
(viii-1) When :
For Case ix, only one type of solution exits; it is where .
For Case x, there are three types of solution with .
(x-1) When :
As an example, we consider solutions of Case (iv-1) when and ; then the solution reduces to the soliton form as for as the standard KdV case and for the standard mKdV case when . We show the diagrams of Case (iv-1) in Figure 1 to illustrate the behaviors of the solutions for different power index . We choose coefficients of the generalized mKdV equation as and ; the constants of the solutions and equal 0.5 and 2, respectively, while the coefficient in the equation equals −0.2. It is found that (i) this kind of solution is a type of soliton solution; the amplitude decreases with the increase of power index ; (ii) it gives that the width of the wave packet broadened when increasing ; (iii) in addition, we find that the velocity of the wave slowed down when is bigger.
Figures of solutions of Case (iv-1). The solution evolutes with spatial coordinate and time . The subscript of indicates the four kinds of different power index of the generalized mKdV equation.
In this paper, we use the modified -expansion method to construct some types of solutions of the mKdV equation through the introduction of a proper transformation. Some new solutions are given, including the hyperbolic, trigonometric, and rational function solutions. It is shown that using the modified )-expansion method we can deal with the nonlinear evolution equations effectively and directly and abundant solutions could be obtained.
The authors acknowledge Qinghua Bu, Qingchun Zhou, and Mingxing Zhu for useful discussion. This work was supported by NSF-China under Grant nos. 11047101, 11105039, 11205071, and 11391240183.
W. Malfliet, “Solitary wave solutions of nonlinear wave equations,” The American Journal of Physics, vol. 60, no. 7, pp. 650–654, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A.-M. Wazwaz, “The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1467–1475, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
F
-expansion to periodic wave solutions for a new Hamiltonian amplitude equation,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1257–1268, 2005. View at: Publisher Site | Google Scholar | MathSciNet
M. A. Abdou, “The extended
F
-expansion method and its application for a class of nonlinear evolution equations,” Chaos, Solitons and Fractals, vol. 31, no. 1, pp. 95–104, 2007. View at: Publisher Site | Google Scholar | MathSciNet
J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” Journal of Mathematical Physics, vol. 24, no. 3, pp. 522–526, 1983. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Liu, Z. Fu, S. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters A, vol. 289, no. 1-2, pp. 69–74, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Hirota, “Exact solution of the korteweg: de vries equation for multiple collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971. View at: Google Scholar
N. Nirmala and M. Vedan, “A variable coefficient Korteweg: de vries equation: similarity analysis and exact solution. II,” Journal of Mathematical Physics, vol. 27, Article ID 2644, 1986. View at: Publisher Site | Google Scholar
M. Wang, Y. Zhou, and Z. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters A, vol. 216, pp. 67–75, 1996. View at: Google Scholar
A. Yıldırım and Z. Pınar, “Application of the exp-function method for solving nonlinear reaction-diffusion equations arising in mathematical biology,” Computers & Mathematics with Applications, vol. 60, no. 7, pp. 1873–1880, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Wang, X. Li, and J. Zhang, “The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008. View at: Publisher Site | Google Scholar | MathSciNet
R. Abazari, “The (G′/G)-expansion method for the coupled Boussinesq equation,” Procedia Engineering, vol. 10, pp. 2845–2850, 2011. View at: Google Scholar
M. Ali Akbar, N. Hj. Mohd. Ali, and E. M. E. Zayed, “Abundant exact traveling wave solutions of generalized Bretherton equation via improved (G′/G)-expansion method,” Communications in Theoretical Physics, vol. 57, no. 2, pp. 173–178, 2012. View at: Publisher Site | Google Scholar | MathSciNet
Y. -L. Zhao, Y. -P. Liu, and Z. -B. Li, “A connection between the (G′/G)-expansion method and the truncated Painlevé expansion method and its application to the mKdV equation,” Chinese Physics B, vol. 19, Article ID 030306, 2010. View at: Google Scholar
I. Aslan, “Exact and explicit solutions to some nonlinear evolution equations by utilizing the (G′/G)-expansion method,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 857–863, 2009. View at: Google Scholar
Y.-B. Zhou and C. Li, “Application of modified (G′/G)-expansion method to traveling wave solutions for Whitham-Broer-Kaup-like equations,” Communications in Theoretical Physics, vol. 51, no. 4, pp. 664–670, 2009. View at: Publisher Site | Google Scholar | MathSciNet
J. Feng, W. Li, and Q. Wan, “Using (G′/G)-expansion method to seek the traveling wave solution of Kolmogorov-Petrovskii-Piskunov equation,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5860–5865, 2011. View at: Publisher Site | Google Scholar | MathSciNet
M. Mizrak and A. Ertaş, “Application of (G′/G)-expansion method to the compound KDV-Burgers-type equations,” Mathematical & Computational Applications, vol. 17, no. 1, pp. 18–28, 2012. View at: Google Scholar | MathSciNet
B. Zheng, “Travelling wave solutions of two nonlinear evolution equations by using the (G′/G)-expansion method,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5743–5753, 2011. View at: Publisher Site | Google Scholar | MathSciNet
M. Wadati, “The modified Korteweg-de Vries equation,” vol. 34, pp. 1289–1296, 1973. View at: Google Scholar | MathSciNet
F. Gesztesy and B. Simon, “Constructing solutions of the mKdV-equation,” Journal of Functional Analysis, vol. 89, no. 1, pp. 53–60, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Kametaka, “Korteweg-de Vries equation. IV. Simplest generalization,” Proceedings of the Japan Academy, vol. 45, pp. 661–665, 1969. View at: Publisher Site | Google Scholar | MathSciNet
S. N. Kruzhkov and A. V. Faminskii, “Generalized solutions of the cauchy problem for the korteweg-de vries equation,” Mathematics of the USSR-Sbornik, vol. 48, no. 2, article 391, 1984. View at: Google Scholar
J. Yang, “Complete eigenfunctions of linearized integrable equations expanded around a soliton solution,” Journal of Mathematical Physics, vol. 41, no. 9, pp. 6614–6638, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. F. Rodríguez, J. A. Reyes, A. Espinosa-Cerón, J. Fujioka, and B. A. Malomed, “Standard and embedded solitons in nematic optical fibers,” Physical Review E, vol. 68, no. 3, Article ID 036606, 2003. View at: Google Scholar
R. Fedele, “Envelope solitons versus solitons,” Physica Scripta, vol. 65, no. 6, article 502, 2002. View at: Publisher Site | Google Scholar
Z. T. Fu, Z. Chen, S. K. Liu, and S. D. Liu, “New solutions to generalized mKdV equation,” Communications in Theoretical Physics, vol. 41, no. 1, pp. 25–28, 2004. View at: Google Scholar
Copyright © 2014 Yu Zhou and Ying Wang. 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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Statistics/Summary/Averages/Harmonic Mean - Wikibooks, open books for an open world
Statistics/Summary/Averages/Harmonic Mean
< Statistics | Summary | Averages
Harmonic Mean[edit | edit source]
The arithmetic mean cannot be used when we want to average quantities such as speed.
Example 1: The distance from my house to town is 40 km. I drove to town at a speed of 40 km per hour and returned home at a speed of 80 km per hour. What was my average speed for the whole trip?.
Solution: If we just took the arithmetic mean of the two speeds I drove at, we would get 60 km per hour. This isn't the correct average speed, however: it ignores the fact that I drove at 40 km per hour for twice as long as I drove at 80 km per hour. To find the correct average speed, we must instead calculate the harmonic mean.
For two quantities A and B, the harmonic mean is given by:
{\displaystyle {\frac {2}{{\frac {1}{A}}+{\frac {1}{B}}}}}
This can be simplified by adding in the denominator and multiplying by the reciprocal:
{\displaystyle {\frac {2}{{\frac {1}{A}}+{\frac {1}{B}}}}={\frac {2}{\frac {B+A}{AB}}}={\frac {2AB}{A+B}}}
For N quantities: A, B, C......
Harmonic mean =
{\displaystyle {\frac {N}{{\frac {1}{A}}+{\frac {1}{B}}+{\frac {1}{C}}+\ldots }}}
Let us try out the formula above on our example:
{\displaystyle {\frac {2AB}{A+B}}}
Our values are A = 40, B = 80. Therefore, harmonic mean
{\displaystyle ={\frac {2\times 40\times 80}{40+80}}={\frac {6400}{120}}\approx 53.333}
Is this result correct? We can verify it. In the example above, the distance between the two towns is 40 km. So the trip from A to B at a speed of 40 km will take 1 hour. The trip from B to A at a speed to 80 km will take 0.5 hours. The total time taken for the round distance (80 km) will be 1.5 hours. The average speed will then be
{\displaystyle {\frac {80}{1.5}}\approx }
53.33 km/hour.
The harmonic mean also has physical significance.
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Quorum (distributed computing) - Wikipedia
(Redirected from Quorum (Distributed Systems))
1 Quorum-based techniques in distributed database systems
1.1 Quorum-based voting in commit protocols
1.2 Quorum-based voting for replica control
Quorum-based techniques in distributed database systems[edit]
Quorum-based voting can be used as a replica control method,[1] as well as a commit method to ensure transaction atomicity in the presence of network partitioning.[1]
Quorum-based voting in commit protocols[edit]
In a distributed database system, a transaction could execute its operations at multiple sites. Since atomicity requires every distributed transaction to be atomic, the transaction must have the same fate (commit or abort) at every site. In case of network partitioning, sites are partitioned and the partitions may not be able to communicate with each other. This is where a quorum-based technique comes in. The fundamental idea is that a transaction is executed if the majority of sites vote to execute it.
Every site in the system is assigned a vote Vi. Let us assume that the total number of votes in the system is V and the abort and commit quorums are Va and Vc, respectively. Then the following rules must be obeyed in the implementation of the commit protocol:
Va + Vc > V, where 0 < Vc, Va
{\displaystyle \leq }
Before a transaction commits, it must obtain a commit quorum Vc.
The total of at least one site that is prepared to commit and zero or more sites waiting
{\displaystyle \geq }
Vc.[2]
Before a transaction aborts, it must obtain an abort quorum Va
The total of zero or more sites that are prepared to abort or any sites waiting
{\displaystyle \geq }
The first rule ensures that a transaction cannot be committed and aborted at the same time. The next two rules indicate the votes that a transaction has to obtain before it can terminate one way or the other.
Quorum-based voting for replica control[edit]
In replicated databases, a data object has copies present at several sites. To ensure serializability, no two transactions should be allowed to read or write a data item concurrently. In case of replicated databases, a quorum-based replica control protocol can be used to ensure that no two copies of a data item are read or written by two transactions concurrently.
The quorum-based voting for replica control is due to [Gifford, 1979].[3] Each copy of a replicated data item is assigned a vote. Each operation then has to obtain a read quorum (Vr) or a write quorum (Vw) to read or write a data item, respectively. If a given data item has a total of V votes, the quorums have to obey the following rules:
The first rule ensures that a data item is not read and written by two transactions concurrently. Additionally, it ensures that a read quorum contains at least one site with the newest version of the data item. The second rule ensures that two write operations from two transactions cannot occur concurrently on the same data item. The two rules ensure that one-copy serializability is maintained.
Replication (computer science)
^ a b Ozsu, Tamer M; Valduriez, Patrick (1991). "12". Principles of distributed database systems (2nd ed.). Upper Saddle River, NJ: Prentice-Hall, Inc. ISBN 978-0-13-691643-7.
^ Skeen, Dale. "A Quorum-based Commit Protocol" (PDF). Cornell University ECommons Library. Retrieved 10 February 2013.
^ Gifford, David K. (1979). Weighted voting for replicated data. SOSP '79: Proceedings of the seventh ACM symposium on Operating systems principles. Pacific Grove, California, United States: ACM. pp. 150–162. doi:10.1145/800215.806583.
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Logarithmic Decrement (δ) | Education Lessons
fig-1: displacement V/s time curve for under damped system
Logarithmic decrement is defined as the natural logarithm of the ratio of successive amplitude on the same side of mean position.
The rate of decay in the amplitudes of under-damped system is measured by the parameter known as logarithmic decrement.
Rate of decay in amplitudes depends on the amount of damping present in the system. So if the damping is more, then the rate of decay will also be more.
Let A and B are the two points on the successive cycles which shows maximum deflection as shown in figure.
The periodic time:
\begin{aligned} t_p &= t_2 − t_1 \\ &= {2 \pi \over \omega_d} \\ &= {2 \pi \over \big(\sqrt {1-\xi^2}\big) \ \omega_n} \end{aligned}
The amplitude at time
t_1
t_2
x_1 = Xe^{-\xi \omega_n t_1} [sin(\omega_d t_1 + \varnothing)]
\begin{aligned} x_2 &= Xe^{-\xi \omega_n t_2} [sin(\omega_d t_2 + \varnothing)] \\ \therefore \quad x_2 &= Xe^{-\xi \omega_n (t_1 + t_p)} [sin \{ \omega_d (t_1 + t_p) + \varnothing \}] \\ \therefore \quad x_2 &= Xe^{-\xi \omega_n (t_1 + t_p)} [sin ( \omega_d t_1 + \omega_d t_p + \varnothing)] \\ \therefore \quad x_2 &= Xe^{-\xi \omega_n (t_1 + t_p)} [sin ( \omega_d t_1 + \omega_d \bigg({2 \pi \over \omega_d}\bigg) + \varnothing)] \\ \therefore \quad x_2 &= Xe^{-\xi \omega_n (t_1 + t_p)} [sin ( \omega_d t_1 + 2 \pi + \varnothing)] \\ \therefore \quad x_2 &= Xe^{-\xi \omega_n (t_1 + t_p)} [sin \{ 2 \pi + (\omega_d t_1 + \varnothing) \}] \\ \therefore \quad x_2 &= Xe^{-\xi \omega_n (t_1 + t_p)} [sin (\omega_d t_1 + \varnothing)] \\ \end{aligned}
Taking ratio, we get;
\begin{aligned} {x_1 \over x_2} &= {Xe^{-\xi \omega_n t_1} [sin(\omega_d t_1 + \varnothing)] \over Xe^{-\xi \omega_n (t_1 + t_p)} [sin (\omega_d t_1 + \varnothing)] }\\ \therefore \quad \quad {x_1 \over x_2} &= e^{-\xi \omega_n (t_1-t_1-t_p)} \\ \therefore \quad \quad {x_1 \over x_2} &= e^{\xi \omega_n t_p} \end{aligned}
The logarithmic decrement is given by;
\begin{aligned} \delta &= log_e \bigg({x_1 \over x_2}\bigg) \\ \therefore \quad \delta &= log_e (e^{\xi \omega_n t_p}) \\ \therefore \quad \delta &= \xi \omega_n t_p \\ \therefore \quad \delta &= \xi \omega_n {2 \pi \over \big(\sqrt {1-\xi^2}\big) \ \omega_n} \\ \therefore \quad \delta &= {2 \pi \xi\over \big(\sqrt {1-\xi^2}\big)} \end{aligned}
The logarithmic decrement can also be determined as follows;
\begin{aligned} \delta &= log_e \bigg({x_0 \over x_1}\bigg)= log_e \bigg({x_1 \over x_2}\bigg)= log_e \bigg({x_2 \over x_3}\bigg) = \dotso = log_e \bigg({x_{n-1} \over x_n}\bigg) \\ \text {Adding upto 'n' terms}\\ n\delta &= log_e \bigg({x_0 \over x_1} \bigg) + log_e \bigg({x_1 \over x_2}\bigg) + log_e \bigg({x_2 \over x_3}\bigg) + \dotso + log_e \bigg({x_{n-1} \over x_n}\bigg) \\ \therefore \ \ n\delta &= log_e \bigg({x_0 \over x_1} \ . {x_1 \over x_2} \ . {x_2 \over x_3} \ . \dots \ . {x_{n-1} \over x_n}\bigg) \\ \text {Or} \qquad \\ n\delta &= log_e \bigg({x_0 \over x_n}\bigg) \\ \therefore \quad \delta &= {1 \over n} log_e \bigg({x_0 \over x_n}\bigg) \end{aligned}
x_0
= amplitude at the starting position
x_n
= amplitude after ‘n’ cycles
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Waves/Beats - Wikibooks, open books for an open world
Waves/Beats
Beats[edit | edit source]
Suppose two sound waves of slightly different frequencies impinge on your ear at the same time. The displacement perceived by your ear is the superposition of these two waves, with time dependence
{\displaystyle A(t)=\sin(\omega _{1}t)+\sin(\omega _{2}t)=2\sin(\omega _{0}t)\cos(\Delta \omega t),}
{\displaystyle \omega _{0}=(\omega _{1}+\omega _{2})/2}
{\displaystyle \Delta \omega =(\omega _{2}-\omega _{1})/2}
. What you actually hear is a tone with angular frequency
{\displaystyle \omega _{0}}
which fades in and out with period
{\displaystyle T_{beat}=\pi /\Delta \omega =2\pi /(\omega _{2}-\omega _{1})=1/(f_{2}-f_{1}).}
The beat frequency is simply
{\displaystyle f_{beat}=1/T_{beat}=f_{2}-f_{1}.}
Note how beats are the time analog of wave packets -- the mathematics are the same except that frequency replaces wavenumber and time replaces space.
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Suppose that the random variables X and Y have joint
Suppose that the random variables X and Y have joint p.d.f.f(x,y)=\begin
f\left(x,y\right)=\left\{\begin{array}{l}kx\left(x-y\right),0<x<2,-x<y<x\\ 0,\text{ }\text{ }\text{ }\text{ }elsewhere\end{array}
f\left(x,y\right)=\left\{\begin{array}{l}kx\left(x-y\right),0<x<2,-x<y<x\\ 0,\text{ }\text{ }\text{ }\text{ }elsewhere\end{array}
{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f\left(x,y\right)dxdy=1
={\int }_{0}^{2}{\int }_{-x}^{x}kx\left(x-y\right)dxdy=1
={\int }_{0}^{2}{\int }_{-x}^{x}k\left({x}^{2}-xy\right)dydx=1
={\int }_{0}^{2}k{\left[{x}^{2}y-\frac{x{y}^{2}}{2}\right]}_{-x}^{x}dx=1
={\int }_{0}^{2}k\left[{x}^{3}-\frac{{x}^{3}}{2}+{x}^{3}+\frac{{x}^{3}}{2}\right]dx=1
={\int }_{0}^{2}k\left[2{x}^{3}\right]dx=1
=2k{\left[\frac{{x}^{4}}{4}\right]}_{0}^{2}=1
=k=\frac{1}{8}
\theta
Until he was in his seventies, Henri LaMothe excited audiences by belly-flopping from a height of 13 m into 35 cm. of water. Assuming that he stops just as he reaches the bottom of the water and estimating his mass to be 72 kg, find the magnitudes of the impulse on him from the water.
Consider a solution
{x}_{1}
of the linear system Ax=b. Justify the facts stated in parts (a) and (b):
{x}_{h}
is a solution of the system Ax=0, then
{x}_{1}+{x}_{h}
is a solution of the system Ax=b.
{x}_{2}
is another solution of the system Ax=b, then
{x}_{2}-{x}_{1}
is a solution of the system Ax=0
Find the inverse Laplace transform
f\left(t\right)={L}^{-1}\left\{F\left(s\right)\right\}
of each of the following functions.
\left(i\right)F\left(s\right)=\frac{2s+1}{{s}^{2}-2s+1}
Hint – Use Partial Fraction Decomposition and the Table of Laplace Transforms.
\left(ii\right)F\left(s\right)=\frac{3s+2}{{s}^{2}-3s+2}
\left(iii\right)F\left(s\right)=\frac{3{s}^{2}+4}{\left({s}^{2}+1\right)\left(s-1\right)}
\sqrt[3]{24}-\sqrt[3]{81}
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Identity Properties of Addition and Multiplication | Brilliant Math & Science Wiki
The identity property of addition is that given some number
Q
Q + 0 = Q .
The identity property of multiplication is that given some number
Z,
Z \times 1 = Z .
Identity properties in general
The identity property of addition in practice
The identity property of multiplication in practice
Identity properties are fundamental to the workings of traditional arithmetic, and in many related systems. They are essentially "operations that do nothing", that leave the identity of a number untouched. This is useful in algebraic manipulation.
Note these properties can apply to number systems other than the real numbers! They are phrased this way here for simplicity.
Identity Property of Addition for Real Numbers: Given any real number
Q,
if 0 is added to
Q,
Q.
Q + 0 = 0 + Q = Q .
Identity Property of Multiplication for Real Numbers: Given any real number
Z,
if 1 is multiplied by
Z,
Z.
Z \times 1 = 1 \times Z = Z .
This particular identity shows up in mental arithmetic. If we have the sum
122 + 59
we can make the arithmetic easier to do by first "adjusting" the 59 so the last digit is 0. Note if we just add 1 we can get 60. However, adding 1 arbitrarily will result in a different number! What we can do instead is add 1 and subtract 1; this is equivalent to adding 0. By applying an identity property, we are guaranteed the number doesn't change.
122 + 59 + 1 - 1 = 122 + 60 - 1
Thus, mentally, we can just 1.) add 122 to 60 and then 2.) adjust by subtracting by 1, which is easier than mentally adding the original ones digits of 2 and 9 and worrying about a carry digit.
Here's an algebra example: suppose you have the expression
x^2 + 6x - 1
and you want to do completing the square. This will rewrite the expression in the format
(x-h)^2 + k
(and makes it easy, for example, to find the vertex of a parabola).
If we think of the expression in the format
ax^2 + bx + c ,
we can take half of the
b
term and square it; the result can be combined in the expression to complete the square. With the example above,
\frac{1}{2} \times 6 = 3 ,
3^2 =9,
so our goal is for the expression to contain
x^2 + 6x + 9 .
However, we can't just change the -1 to a 9; what we can do is add 9 and subtract 9, which is equivalent to adding 0. Since adding 0 is an identity property, nothing is changed.
x^2 + 6x + 9 - 9 - 1
Now we can finish completing the square, and combine
-9 - 1
as a separate term.
(x+3)^2 - 10
If we are dealing with an equation, note an alternative to using the addition property of identity would be to add the same value to both sides of the equal sign. For example, given
3 = x^2 + 6x - 1 ,
we could add 9 to both sides and get
3 + 9 = x^2 + 6x + 9 - 1 .
However, sometimes the expression being worked with is embedded in a larger equation and using both sides of the equal sign isn't possible; in such a scenario the addition property of identity is completely necessary.
2 c
1
c^2
0
a - (a - b - a - c + a) - b + c?
Simplifying and reducing fractions often applies this identity. For example, suppose you have the fraction
\frac{4}{25}
but you want a denominator of 100 (perhaps in making a percentage). You want to multiply the 25 by 4, turning it into 100. However, you can't just arbitrarily do that, because it changes the number. What you can do is multiply by
\frac{4}{4} ,
which is equivalent to multiplying by 1. The identity property of multiplication means that this won't change the number:
\frac{4}{25} \times \frac{4}{4} = \frac{16}{100} .
\frac{841}{840}
\frac{ 211}{200}
\frac{ 9}{8}
\left ( 1 + \frac{1}{ \color{#D61F06}{4}} \right ) \left ( 1 + \frac{1}{\color{#D61F06}{5}} \right ) \left ( 1 + \frac{1}{\color{#D61F06}{6}} \right ) \left ( 1 + \frac{1}{\color{#D61F06}{7}} \right )
The algebraic analogue of fractions is rational expressions, where the same sort of logic applies.
\frac{2x}{3x^2} + \frac{x}{3x}
To do the addition with the example above, we'd like to have both denominators read
3x^2 .
However, the second term
\frac{x}{3x}
is missing an
x
in the denominator. We can't just multiply it in without changing the number, but we can apply the identity property of multiplication and use
\frac{x}{x}
(Warning note:
\frac{x}{x}
is not 1 if
x
is 0. Since the current expression already is invalid when
x=0,
this does not change the situation here.)
\frac{2x}{3x^2} + \frac{x}{3x} \frac{x}{x} = \frac{2x}{3x^2} + \frac{x^2}{3x^2}
Now that the denominators are the same, we can add and get
\frac{2x+x^2}{3x^2} .
Cite as: Identity Properties of Addition and Multiplication. Brilliant.org. Retrieved from https://brilliant.org/wiki/identity-properties-of-addition-and-multiplication/
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Laurent polynomials associated with wavelet - MATLAB wave2lp - MathWorks 日本
wave2lp
Laurent Polynomials Associated with Wavelet
PmaxHS
AddPOW
PRCond,AACond
wave2lp input syntax has changed
[LoDz,HiDz,LoRz,HiRz] = wave2lp(wname)
[___,PRCond,AACond] = wave2lp(wname)
[___] = wave2lp(wname,PmaxHS)
[___] = wave2lp(wname,PmaxHS,AddPOW)
[LoDz,HiDz,LoRz,HiRz] = wave2lp(wname) returns the four Laurent polynomials associated with the wavelet wname. The pairs (LoRz,HiRz) and (LoDz,HiDz) are associated with the synthesis and analysis filters, respectively.
[___,PRCond,AACond] = wave2lp(wname) also returns the perfect reconstruction condition PRCond and the anti-aliasing condition AACond.
[___] = wave2lp(wname,PmaxHS) sets the maximum order of LoRz.
[___] = wave2lp(wname,PmaxHS,AddPOW) sets the maximum order of the Laurent polynomial HiRz.
Obtain the four Laurent polynomials associated with the orthogonal wavelet db3. Also obtain the perfect reconstruction and anti-aliasing conditions.
[LoDz,HiDz,LoRz,HiRz,PRC,AAC] = wave2lp("db3")
LoDz =
Coefficients: [0.0352 -0.0854 -0.1350 0.4599 0.8069 0.3327]
HiDz =
Coefficients: [0.3327 -0.8069 0.4599 0.1350 -0.0854 -0.0352]
LoRz =
Coefficients: [0.3327 0.8069 0.4599 -0.1350 -0.0854 0.0352]
HiRz =
Coefficients: [-0.0352 -0.0854 0.1350 0.4599 -0.8069 0.3327]
PRC =
AAC =
Verify the perfect reconstruction condition.
eq(LoRz*LoDz + HiRz*HiDz,PRC)
Verify the anti-aliasing condition. Use the helper function helperMakeLaurentPoly to obtain
LoD\left(-z\right)
LoD\left(z\right)
is the Laurent polynomial LoDz. Use the helper function helperMakeLaurentPoly to obtain
HiD\left(-z\right)
HiD\left(z\right)
is the Laurent polynomial HiDz.
LoDzm = helperMakeLaurentPoly(LoDz);
HiDzm = helperMakeLaurentPoly(HiDz);
eq(LoRz*LoDzm + HiRz*HiDzm,AAC)
function polyout = helperMakeLaurentPoly(poly)
% This function is only intended to support this example.
polyout = poly;
cflen = length(polyout.Coefficients);
cmo = polyout.MaxOrder;
polyneg = (-1).^(mod(cmo,2)+(0:cflen-1));
polyout.Coefficients = polyout.Coefficients.*polyneg;
Wavelet, specified as a character vector or string scalar. wname must be one of the wavelets supported by liftingScheme. See the Wavelet property of liftingScheme for the list of wavelets.
Example: [LoDz,HiDz,LoRz,HiRz] = wave2lp("db2")
PmaxHS — Maximum power
Maximum power of the Laurent polynomial LoRz, specified as an integer.
Example: If [~,~,LoRz,HiRz] = wave2lp("db2",3), then the maximum power, or order, of the Laurent polynomial LoRz is 3.
AddPOW — Integer
Integer to set the maximum order of the Laurent polynomial HiRz. PmaxHiRz, the maximum order of HiRz, is
PmaxHiRz = PmaxHS+length(HiRz.Coefficients)-2+AddPow.
AddPOW must be an even integer to preserve the perfect reconstruction condition.
LoDz — Laurent polynomial
laurentPolynomial object
Laurent polynomial associated with the lowpass analysis filter, returned as a laurentPolynomial object.
HiDz — Laurent polynomial
Laurent polynomial associated with the highpass analysis filter, returned as a laurentPolynomial object.
LoRz — Laurent polynomial
Laurent polynomial associated with the lowpass synthesis filter, returned as a laurentPolynomial object.
HiRz — Laurent polynomial
Laurent polynomial associated with the highpass synthesis filter, returned as a laurentPolynomial object.
PRCond,AACond — Perfect reconstruction and anti-aliasing conditions
laurentPolynomial objects
Perfect reconstruction and anti-aliasing conditions, returned as laurentPolynomial objects. The perfect reconstruction condition PRCond and anti-aliasing condition AACond are:
PRCond(z) = LoRz(z) LoDz(z) + HiRz(z) HiDz(z)
AACond(z) = LoRz(z) LoDz(-z) + HiRz(z) HiDz(-z)
The pairs (LoRz, HiRz) and (LoDz, HiDz) are associated with perfect reconstructions filters if and only if:
PRCond(z) = 2, and
AACond(z) = 0
If PRCond(z) = 2 zd, a delay is introduced in the reconstruction process.
R2021b: wave2lp input syntax has changed
The wave2lp input syntax has changed.
You can now set the maximum order of LoRz using PmaxHS.
You can now set the maximum order of HiRz using AddPOW.
filters2lp | lp2filters
laurentMatrix | laurentPolynomial | liftingScheme
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LMIs in Control/pages/LMI for Attitude Control of Nonrotating Missiles - Wikibooks, open books for an open world
LMIs in Control/pages/LMI for Attitude Control of Nonrotating Missiles
The dynamic model of a missile is very complicated and a simplified model is used. To do so, we consider a simplified attitude system model for the pitch channel in the system. We aim to achieve a non-rotating motion of missiles. It is worthwhile to note that the attitude control design for the pitch channel and the yaw/roll channel can be solved exactly in the same way while representing matrices of the system are different.
{\displaystyle {\begin{aligned}{\dot {x}}(t)&=A(t)x(t)+B_{1}(t)u(t)+B_{2}(t)d(t)\\y(t)&=C(t)x(t)+D_{1}(t)u(t)+D_{2}(t)d(t)\end{aligned}}}
{\displaystyle x=[\alpha \quad w_{z}\quad \delta _{z}]^{\text{T}}}
{\displaystyle u=\delta _{zc}}
{\displaystyle y=[\alpha \quad n_{y}]^{\text{T}}}
{\displaystyle d=[\beta \quad w_{y}]^{\text{T}}}
{\displaystyle \alpha }
{\displaystyle w_{z}}
{\displaystyle \delta _{z}}
{\displaystyle \delta _{zc}}
{\displaystyle n_{y}}
{\displaystyle \beta }
{\displaystyle w_{y}}
In the aforementioned pitch channel system, the matrices
{\displaystyle A(t),B_{1}(t),B_{2}(t),C(t),D_{1}(t),}
{\displaystyle D_{2}(t)}
{\displaystyle {\begin{aligned}A(t)={\begin{bmatrix}-a_{4}(t)&1&-a_{5}(t)\\-{\acute {a}}_{1}(t)a_{4}(t)-a_{2}(t)&{\acute {a}}_{1}(t)-a_{1}(t)&{\acute {a}}_{1}(t)a_{5}(t)-a_{3}(t)\\0&0&-1/\tau _{z}\end{bmatrix}}\end{aligned}}}
{\displaystyle {\begin{aligned}B_{1}(t)={\begin{bmatrix}0\\0\\1\end{bmatrix}},\quad B_{2}(t)={\frac {w_{x}}{57.3}}{\begin{bmatrix}-1&0\\-{\acute {a}}_{1}(t)&{\frac {J_{x}-J_{y}}{J_{z}}}\\0&0\end{bmatrix}}\end{aligned}}}
{\displaystyle {\begin{aligned}C(t)={\frac {w_{x}}{57.3}}{\begin{bmatrix}57.3g&0&0\\V(t)a_{4}(t)&0&V(t)a_{5}(t)\end{bmatrix}}\end{aligned}}}
{\displaystyle {\begin{aligned}D_{1}(t)=0,\quad D_{2}(t)={\frac {1}{57.3g}}{\begin{bmatrix}0&0\\V(t)b_{7}(t)&0\end{bmatrix}}\end{aligned}}}
{\displaystyle a_{1}(t)\sim a_{6}(t),\quad b_{1}(t)\sim b_{7}(t),{\acute {a}}_{1}(t),{\acute {b}}_{1}(t)}
{\displaystyle c_{1}(t)\sim c_{4}(t)}
{\displaystyle V}
{\displaystyle J_{x}}
{\displaystyle J_{y}}
{\displaystyle J_{z}}
{\displaystyle u=Kx}
{\displaystyle {\begin{aligned}{\dot {x}}&=(A+B_{1}K)x+B_{2}d\\z&=(C+D_{1}K)x+D_{2}d\end{aligned}}}
{\displaystyle H_{\infty }}
{\displaystyle G_{zd}(s)=(C+D_{1}K)(sI-(A+B_{1}K))^{-1}B_{2}+D_{2}}
{\displaystyle \gamma }
{\displaystyle ||G_{zd}(s)||_{\infty }<\gamma }
{\displaystyle {\begin{aligned}&{\text{min}}\quad \gamma \\&{\text{s.t.}}\quad X>0\\&{\begin{bmatrix}(AX+B_{1}W)^{T}+AX+B_{1}W&B_{2}&(CX+D_{1}W)^{T}\\B_{2}^{T}&-\gamma I&D_{2}^{T}\\CX+D_{1}W&D_{2}&-\gamma I\end{bmatrix}}<0\end{aligned}}}
{\displaystyle \gamma }
{\displaystyle W}
{\displaystyle X}
{\displaystyle K=WX^{-1}}
https://github.com/asalimil/LMI-for-Non-rotating-Missle-Attitude-Control
Retrieved from "https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/pages/LMI_for_Attitude_Control_of_Nonrotating_Missiles&oldid=3794630"
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The exponential growth models describe the population of the indicated
The exponential growth models describe the population of the indicated country,
The exponential growth models describe the population of the indicated country, A, in millions, t years after 2006.Uganda’s growth rate is approximately 3.8 times that of Canada’s.Determine whether the statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
A=33.1{e}^{0.009t}
A=28.2{e}^{0.0034t}
The exponential growth rate of population of Canada is
A=33.1{e}^{0.009t}
The exponential growth rate of population of Uganda is
A=28.2{e}^{0.034t}
The exponential constant of the exponential growth rate of population of Canada is
{k}_{c}=0.009
The exponential constant of the exponential growth rate of population of Uganda is
{k}_{u}=0.034
The expression for the ratio of the exponential constant of the exponential growth rate of population of Canada to exponential constant of the exponential growth rate of population of Uganda is,
\frac{{k}_{u}}{{k}_{c}}=\frac{0.034}{0.009}
{k}_{u}=3.777{k}_{c}
\approx 3.80{k}_{c}
Hence the statement is true that Uganda’s growth rate is approximately 3.8 times that of Canada’s.
A=1173.1{e}^{0.008t}
A=31.5{e}^{0.019t}
A=127.3{e}^{0.006t}
A=141.9{e}^{0.005t}
\begin{array}{|cccccc|}\hline x& -1& 0& 1& 2& 3\\ g\left(x\right)& 2& 5& 8& 11& 14\\ \hline\end{array}
Let V denote rainfall volume and W denote runoff volume (both in mm). According to the article “Runoff Quality Analysis of Urban Catchments with Analytical Probability Models” (J. of Water Resource Planning and Management, 2006: 4–14), the runoff volume will be 0 if
\left[V\text{ }\le \text{ }{v}_{d}\text{ }\right]
and will
\left[k\text{ }\left(V\text{ }-\text{ }{v}_{d}\right)\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}\text{ }V\text{ }>\text{ }{v}_{d}.\text{ }Here\text{ }{v}_{d}\right]
is the volume of depression storage (a constant) and k (also a constant) is the runoff coefficient. The cited article proposes an exponential distribution with parameter
\left[\lambda \text{ }f\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\text{ }V.\right]
[Note: W is neither purely continuous nor purely discrete, instead it has a “mixed” distribution with a discrete component at 0 and is continuous for values
w>0
b. What is the pdf of W for
w>0
? Use this to obtain an expression for the expected value of runoff volume.
The number of users on a website has grown exponentially since its launch. After 2 months, there were 300 users. After 4 months there were 30000 users. Find the exponential function that models the number of users x months after the website was launched.
The population of a small town was 3,600 people in the year 2015. The population increases by 4.5% every year.
Write the exponential equation that models the population of the town t years after 2015.
Then use your equation to estimate the population in the year 2025?
A computer valued at $1500 loses 20% of its value each year. a. Write a function rule that models the value of the computer. b. Find the value of the computer after 3 yr. c. In how many years will the value of the computer be less than $500? Remember the general form of an exponential function:
f\left(x\right)=a\ast {b}^{x}
. Given your table and situation, determine values for a and b. Write the appropriate exponential function.
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find domain and range of (x2-9)/x-3 - Maths - Relations and Functions - 8206827 | Meritnation.com
f\left(x\right)=\frac{{x}^{2}-9}{x-3}
since the denominator can not be zero.
x-3\ne 0\phantom{\rule{0ex}{0ex}}⇒x\ne 3
the domain of the function is
R-\left\{3\right\}
y=f\left(x\right)=\frac{{x}^{2}-9}{x-3}\phantom{\rule{0ex}{0ex}}y=\frac{\left(x-3\right)\left(x+3\right)}{x-3}\phantom{\rule{0ex}{0ex}}y=x+3 for x\in R-\left\{3\right\}\phantom{\rule{0ex}{0ex}}y\ne 3+3 \left[f\left(x\right) is not defined for x=3\phantom{\rule{0ex}{0ex}}y\ne 6
range of the function is
R-\left\{6\right\}
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Weighting function with monotonic gain profile - MATLAB makeweight - MathWorks í•œêµ
\begin{array}{c}W\left(0\right)=\text{dcgain}\\ W\left(\text{Inf}\right)=\text{hfgain}\\ |W\left(jâ
\text{freq}\right)|=\text{mag}\text{.}\end{array}
\begin{array}{c}W\left(1\right)=\text{dcgain}\\ W\left(â1\right)=\text{hfgain}\\ |W\left({e}^{jâ
\text{freq}â
\text{Ts}}\right)|=\text{mag}\text{.}\end{array}
\begin{array}{c}W\left(0\right)=\text{dcgain}\\ W\left(\text{Inf}\right)=\text{hfgain}\\ |W\left(jâ
\text{freq}\right)|=\text{mag}\text{.}\end{array}
\begin{array}{c}W\left(1\right)=\text{dcgain}\\ W\left(â1\right)=\text{hfgain}\\ |W\left({e}^{jâ
\text{freq}â
\text{Ts}}\right)|=\text{mag}\text{.}\end{array}
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Let's say you're perusing the want ads and come upon an ad for an equity analyst. The pay is great; there are travel opportunities. It looks like the job for you. Glancing down the list of qualifications, you mentally check off each one:
Bachelor's in engineering or mathematics - check
Master's in economics or business administration - check
Curious, creative thinker - check
Can interpret financial statements - check
Strong technical analytical skills - check
Modeling experience required - check, no wait, better get some 8x10 glossies made up.
The truth is, when companies want their equity analysts to have modeling experience they don't care how photogenic they are. What the term refers to is an important and complicated part of equity analysis known as financial modeling. In this article, we'll explore what a financial model is and how to create one.
Theoretically, a financial model is a set of assumptions about future business conditions that drive projections of a company's revenue, earnings, cash flows, and balance sheet accounts.
In practice, a financial model is a spreadsheet (usually in Microsoft's Excel software) that analysts use to forecast a company's future financial performance. Properly projecting earnings and cash flows into the future is important since the intrinsic value of a stock depends largely on the outlook for financial performance of the issuing company.
A financial model spreadsheet usually looks like a table of financial data organized into fiscal quarters and/or years. Each column of the table represents the balance sheet, income statement, and cash flow statement of a future quarter or year. The rows of the table represent all the line items of the company's financial statements, such as revenue, expenses, share count, capital expenditures and balance sheet accounts. Like financial statements, one generally reads the model from the top to the bottom or revenue through earnings and cash flows.
Each quarter embeds a set of assumptions for that period, like the revenue growth rate, the gross margin assumption, and the expected tax rate. These assumptions are what drive the output of the model - generally, earnings and cash flow figures that are used to value the company or help in making financing decisions for the company.
When trying to predict the future, a good place to start is the past. Therefore, a good first step in building a model is to fully analyze a set of historical financial data and link projections to the historical data as a base for the model. If a company has generated gross margins in the 40% to 45% range for the past ten years, then it might be acceptable to assume that, with other things being equal, a margin of this level is sustainable into the future.
Consequently, the historical track record of gross margin can become somewhat of a basis for a future income projection. Analysts are always smart to examine and analyze historical trends in revenue growth, expenses, capital expenditures, and other financial metrics before attempting to project financial results into the future. For this reason, financial model spreadsheets usually incorporate a set of historical financial data and related analytical measures from which analysts derive assumptions and projections.
Revenue growth rate assumptions can be one of the most important assumptions in a financial model. Small variances in top-line growth can mean big variances in earnings per share (EPS) and cash flows and therefore stock valuation. For this reason, analysts must pay a lot of attention to getting the top-line projection right. A good starting point is to look at the historic track record of revenue. Perhaps revenue is stable from year to year. Perhaps it is sensitive to changes in national income or other economic variables over time. Perhaps growth is accelerating, or maybe the opposite is true. It is important to get a feel for what has affected revenue in the past in order to make a good assumption about the future.
Once one has examined the historic trend, including what's been going on in the most recently reported quarters, it is wise to check if management has given revenue guidance, which is management's own outlook for the future. From there analyze if the outlook is reasonably conservative, or optimistic based on a thorough analytical overview of the business.
A future quarter's revenue projection is frequently driven by a formula in the worksheet such as:
\begin{aligned} &R_1=R_0 \times (1 + g) \\ &\textbf{where:}\\ &R_1=\text{future revenue}\\ &R_0=\text{current revenue}\\ &g=\text{percentage growth rate}\\ \end{aligned}
R1=R0×(1+g)where:R1=future revenueR0=current revenueg=percentage growth rate
Operating Expenses and Margin
Again, the historic trend is a good place to start when forecasting expenses. Acknowledging that there are big differences between the fixed costs and variable costs incurred by a business, analysts are smart to consider both the dollar amount of costs and their proportion of revenue over time. If selling, general and administrative (SG&A) expense has ranged between 8% and 10% of revenue in the past ten years, then it is likely to fall into that range in the future. This could be the basis for a projection - again tempered by management's guidance and an outlook for the business as a whole. If business is improving rapidly, reflected by the revenue growth assumption, then perhaps the fixed cost element of SG&A will be spread over a larger revenue base and the SG&A expense proportion will be smaller next year than it is right now. That means that margins are likely to increase, which could be a good sign for equity investors.
Expense-line assumptions are often reflected as percentages of revenue and the spreadsheet cells containing expense items usually have formulas such as:
\begin{aligned} &E_1=R_1 \times p \\ &\textbf{where:}\\ &E_1=\text{expense}\\ &R_1=\text{revenue for the period}\\ &p=\text{expense percentage of revenue for the period}\\ \end{aligned}
E1=R1×pwhere:E1=expenseR1=revenue for the periodp=expense percentage of revenue for the period
For an industrial company, non-operating expenses are primarily interest expense and income taxes. The important thing to remember when projecting interest expense is that it is a proportion of debt and is not explicitly tied to operational income streams. An important analytical consideration is the current level of total debt owed by the company. Taxes are generally not linked to revenue, but rather pre-tax income. The tax rate that a company pays can be affected by a number of factors such as the number of countries in which it operates. If a company is purely domestic, then an analyst might be safe using the state tax rate as a good assumption in projections. Once again, it is useful to look at the historic track record in these line items as a guide for the future.
Earnings and Earnings Per Share
Projected net income available for common shareholders is projected revenue minus projected expenses.
Projected earnings per share (EPS) is this figure divided by the projected fully diluted shares outstanding figure. Earnings and EPS projections are generally considered primary outcomes of a financial model because they are frequently used to value equities or generate target prices for a stock.
To calculate a one-year target price, the analyst can simply look to the model to find the EPS figure for four quarters in the future and multiply it by an assumed P/E multiple. The projected return from the stock (excluding dividends) is the percentage difference from that target price to the current price:
\begin{aligned} &\text{Projected return}=\frac{(T-P)}{T} \\ &\textbf{where:}\\ &T=\text{target price}\\ &P=\text{current price}\\ \end{aligned}
Projected return=T(T−P)where:T=target priceP=current price
Now the analyst has a simple basis for making an investment decision - the expected return on the stock.
Since the present value of a stock is inextricably linked to the outlook for financial performance of the issuer, investors are wise to create some form of financial projection to evaluate equity investments. Examining the past in an analytical context is only half the story (or less). Developing an understanding of how a company's financial statements might look in the future is often the key to equity valuation.
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Static and Dynamic Balancing | Education Lessons
When machine is in working condition different forces are acting on it which may cause machine to vibrate and cause damage to machine parts. The different forces acting on machine parts are static forces and dynamic forces.
The force which depends on weight of a body, is known as static force. (Generally, a static force acts when vibrations occurs in same plane.)
Dynamic or Inertia force
The force which depends on acceleration of a body, is known as dynamic force. (Generally, dynamic force acts when vibrations occurs in different planes.)
Due to these forces, the efficiency of the system decreases and life span of the system also decreases.
Due to these forces, the machine starts vibrating and sometimes when the vibrations increases, the machine would lift from it’s position and cause damage to other machine or human. So to avoid this, foundation is made below the machine (as you can see in the Fig-1 below), which absorbs the vibration and protects the machine against causing damage. Thus, balancing of the machine is required.
Balancing is the process of eliminating, the effect of static forces and dynamic forces acting on machine components.
When is system said to be unbalanced?
In any system with one or more rotating masses, if the centre of mass of the system does not lie on the axis of rotation, then the system is said to be unbalanced.
As you can see in the Fig-2 we have a rotor which is mounted on a shaft and the shaft has its own axis of rotation. Now you can see that the C.G(centre of gravity) of the rotor is at a distance r from the axis of rotation of the shaft, so when the rotor will start to rotate, a centrifugal force will act on it which is in the outward direction as you can see in Fig-3. Due to this force our system will become unbalanced and it will start to vibrate.
A system is said to be statically balanced, if the centre of masses(C.G) of the system lies on the axis of rotation.
Condition for Static Balancing
The resultant of all the centrifugal forces (dynamic forces) acting on the system during rotation must be zero.
\begin{aligned} \small ∑ \text {Centrifugal forces acting on the system} &= \text {zero}\\ i.e. \ +mrω^2 – mrω^2 &= 0 \end{aligned}
A system is said to be dynamically balanced, if it satisfies following two conditions:-
The resultant of all the dynamic forces acting on the system during rotation must be zero.
∑ \text {Dynamic forces acting on the system} = \text {zero}
The resultant couple due to all the dynamic forces acting on the system during rotation, about any plane, must be zero.
∑ \text {couple} = \text {zero}
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Retention Of Cardiac Auscultation Skill Requires Paired Visual And Audio Information In New Learners | J. Med. Devices | ASME Digital Collection
Glenn Nordehn,
Univ. of MN Med. School Duluth
, MED 153, 1035 Univ. Drive, Duluth, MN 55812
Spencer Strunic,
Spencer Strunic
Tom Soldner,
Tom Soldner
Nicholas Karlisch,
Nicholas Karlisch
Ian Kramer,
Nordehn, G., Strunic, S., Soldner, T., Karlisch, N., Kramer, I., and Burns, S. (June 10, 2008). "Retention Of Cardiac Auscultation Skill Requires Paired Visual And Audio Information In New Learners." ASME. J. Med. Devices. June 2008; 2(2): 027503. https://doi.org/10.1115/1.2927390
Introduction: Cardiac auscultation accuracy is poor: 20% to 40%. Audio-only of 500 heart sounds cycles over a short time period significantly improved auscultation scores. Hypothesis: adding visual information to an audio-only format, significantly
(p<.05)
improves short and long term accuracy. Methods: Pre-test: Twenty-two 1st and 2nd year medical student participants took an audio-only pre-test. Seven students comprising our audio-only training cohort heard audio-only, of 500 heart sound repetitions. 15 students comprising our paired visual with audio cohort heard and simultaneously watched video spectrograms of the heart sounds. Immediately after trainings, both cohorts took audio-only post-tests; the visual with audio cohort also took a visual with audio post-test, a test providing audio with simultaneous video spectrograms. All tests were repeated in six months. Results: All tests given immediately after trainings showed significant improvement with no significant difference between the cohorts. Six months later neither cohorts maintained significant improvement on audio-only post-tests. Six months later the visual with audio cohort maintained significant improvement
(p<.05)
on the visual with audio post-test. Conclusions: Audio retention of heart sound recognition is not maintained if: trained using audio-only; or, trained using visual with audio. Providing visual with audio in training and testing allows retention of auscultation accuracy. Devices providing visual information during auscultation could prove beneficial.
bioacoustics, cardiology, patient diagnosis
Biomedicine, Cardiology, Cycles, Patient diagnosis, Students, Testing
Impedimetric Feedback in Particle Laden Digital Microfluidic Devices
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how to draw the graph of mod x-1 - Maths - Relations and Functions - 9740979 | Meritnation.com
\mathrm{Note} \mathrm{that} \mathrm{the} \mathrm{mode} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right)=\left|\mathrm{x}\right| \mathrm{is} \mathrm{defined} \mathrm{as},\phantom{\rule{0ex}{0ex}}
\mathrm{In} \mathrm{a} \mathrm{similar} \mathrm{way}, \mathrm{the} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right)=\left|\mathrm{x}-1\right| \mathrm{can} \mathrm{be} \mathrm{defined} \mathrm{as} \mathrm{follows}:\phantom{\rule{0ex}{0ex}}
\mathrm{So} \mathrm{the} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right)=\left|\mathrm{x}-1\right| \mathrm{is} \mathrm{a} \mathrm{piecewise} \mathrm{function} \mathrm{taking} \mathrm{seperate} \mathrm{values} \mathrm{on} \mathrm{the} \phantom{\rule{0ex}{0ex}}\mathrm{left} \mathrm{and} \mathrm{right} \mathrm{of} \mathrm{the} \mathrm{point} \mathrm{x}=1.\phantom{\rule{0ex}{0ex}}\mathrm{To} \mathrm{plot} \mathrm{the} \mathrm{graph} \mathrm{of} \mathrm{this} \mathrm{function}, \mathrm{plot} \mathrm{the} \mathrm{graph} \mathrm{of} \mathrm{the} \mathrm{equations},\phantom{\rule{0ex}{0ex}} \mathrm{y}=-\mathrm{x}+1 \mathrm{for} \mathrm{x}<1\phantom{\rule{0ex}{0ex}} \mathrm{y}=\mathrm{x}-1 \mathrm{for} \mathrm{x}\ge 1\phantom{\rule{0ex}{0ex}}\mathrm{The} \mathrm{graph} \mathrm{of} \mathrm{the} \mathrm{function} \mathrm{will} \mathrm{be} \mathrm{V} \mathrm{shaped} \mathrm{at} \mathrm{x}=1.\phantom{\rule{0ex}{0ex}}\mathrm{The} \mathrm{graph} \mathrm{is} \mathrm{shown} \mathrm{below}:\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}} \phantom{\rule{0ex}{0ex}}
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How do you use algebra to solve (x-1)(x+2)=18?
Arely Briggs 2022-01-23 Answered
How do you use algebra to solve
\left(x-1\right)\left(x+2\right)=18
enveradapb
We know that a if a product is 0, then at least one of the factors must be 0. But, thats
14{n}^{2}+42n-2520=0
for n.
Let f(x) be a quadratic function of the form
{x}^{2}+px+q
f\left(f\left(x\right)\right)=0
has two equal real roots. Show that
p\ge 0\text{ }\text{and}\text{ }q\ge 0
y=3{x}^{3}+7{x}^{2}-48x+49
and that y has the same remainder when it is divided by
x+k
x-k
, find the possible values of k.
{x}^{2}-2xy+6{y}^{2}-12x+2y+41\ge 0
where x,
y\in \mathbb{R}
A quadratic has an axis of symmetry of x=m, and the quadratic hax an x-intercept at (m,0). The quadratic has a vertical compression as a approaches infinty y approaches negative infiniy. What is a possible equation of the quadratic?
{x}^{2}+2x=k+{\int }_{0}^{1}|t+k|dt
Then choose the correct option(s),
A) Roots are Real
B) Roots are Imaginary
C) Roots are Distinct
D) Roots are complex number
Solving the quadratic equation
a{X}^{2}+a{Y}^{2}+2bX+2cY+d=0
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Algebra/Intercepts - Wikibooks, open books for an open world
Algebra/Intercepts
← Linear Equations and Functions Intercepts Slope →
Intercepts[edit | edit source]
To find where the equation of a line crosses the X or Y axis, you don't need much information. In this section we will look at how to find the where the line crosses the axes using the standard form for the linear equation. After we look at how slope works we will see we can convert between the various types of linear equations into the standard form.
X and Y axis intercepts[edit | edit source]
An axis intercept point is a point where the graph of a function, relation, or equation intersects the X or Y axes. This section is about finding out how a particular set of functions: linear functions cross the axes.
We know that the domains of most lines are infinite because they are defined at every value of X. The exception is lines that are defined as
{\displaystyle X=c}
where c is a number we choose when we write the function. By definition this line is only defined for one value of X. Since the domain maps onto more than one value for the range this is actually a relationship and not a function. We've seen the graph of this relationships is a vertical line that passes through the point (c,Y). In the picture below we show that when c=0 then our line is the same as the Y axis. When
{\displaystyle c\neq 0}
(as in the drawing where X = 3) then the line can never intercept the Y axis.
Lines with the equation X=C intersect the X axis once and the Y axis 0 or infinite times.
We can also restrict the range of a function by simply writing
{\displaystyle Y=c}
where c is again any number we choose. The graph of this line is a horizontal line that passes through the point (X,c). When Y=o this line is the same as the X axis. when
{\displaystyle c\neq 0}
(as in the drawing where Y = 3)then the line can never intercept the X axis.
Lines with the equation Y=C intersect the Y axis once and the X axis 0 or infinite times.
When looking at Cartesian graphs and linear equations we run into a mathematical axiom: "Two points determine a line.". We will see how this axiom affects the slope-intercept definition of a line
{\displaystyle y=f(x)=mx+b}
in the next section. When two lines intersect they intersect at a point. If a line is not horizontal or perpendicular it will have to intersect the X and Y axes once, but only once.
In this book we are going to accept the statement "At most one line can be drawn through any point not on a given line parallel to the given line in a plane." There is a branch of mathematics called "non-euclidean geometry" that was founded a little more than 160 years ago. Even if you are not interested in mathematics it is worth looking at this Wikipedia article on geometry to get a feel for how formalizing geometry with algebraic methods and then moving beyond them has changed civilization. If you continue in a career requiring advanced mathematics such as Engineering or Physics you might want to follow your interests to see the effect of non-euclidean geometry in your career.
We've seen that for the equation Y=mX + b the Y intercept will always be at b because that is where X=0.
Using Algebra we can subtract b from both sides: Y - b = mX
and multiply by
{\displaystyle {\frac {1}{m}}}
{\displaystyle {\frac {Y}{m}}-{\frac {b}{m}}=X}
we can see that the X intercept is going to be
{\displaystyle -{\frac {b}{m}}}
An axis intercept may simply refer to the number value on the axis where the intersection occurs. For brevity we may say the line has an X intercept of 1 and a Y intercept of 2. After graphing just a few lines you will be able to tell this line points down and runs through quadrants II, I, and IV. With a little more practice you will be able to know that the equation for the line is Y=-2x + 2. We will see that by specifying the two points we are actually implying the slope of the line. There is an exception to this rule. If we say a line crosses the axes at 0 we know that the line will pass through 2 quadrants instead of 3, but we won't know which quadrants or how steep the line is. When we look at slope in the next section we will see why the equations above specify a point and a slope.
When you are trying to graph a linear equation finding the axes intercepts is often the easiest way to go about doing it. To find the x-intercept, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y. For most examples the intercepts are different points, and a line can be drawn through the two intercepts. If both intercepts are (0,0), then another point must be determined to graph the line. If the equations is in the form x = c or y = c, the horizontal or vertical lines are very simple to plot.
Need example graphs showing lines
Need problems
{\displaystyle Y=5\times x+2}
{\displaystyle Y=5\times 0+2}
Substitute zero for x
{\displaystyle Y=2}
Therefore, the Y-Intercept of Y = 5x + 2 is 2.
This works for any form of equation.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Algebra/Intercepts&oldid=3251291"
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Conic Sections, Studymaterial: CBSE Class 11-commerce SCIENCE, Science - Meritnation
Conic sections or conics are the curves obtained by intersecting a double-napped right-circular cone with a plane.
The concept of conic sections is widely used in astronomy, projectile motion of an object, etc.
The example of conic sections are circle (Figure I), ellipse (Figure II), parabola (Figure III) and hyperbola (Figure IV).
Different types of conics can be formed by intersecting a plane with a double-napped cone (other than the vertex) by different ways.
If θ1 is the angle between the axis and the generator and θ2 is the angle between the plane and the axis, then for different conditions of θ1 and θ2, we get different conics. These are described in the table shown below.
θ2 = 90° (The plane cuts only one nappe of the cone entirely)
θ1 < θ2 < 90° (The plane cuts only one nappe of the cone entirely)
θ1 = θ2 (The plane cuts only one nappe of the cone entirely)
0 ≤ θ2 < θ1 (The plane cuts each nappe of the cone entirely)
The conic sections obtained by cutting a plane with a double-napped cone at its vertex are known as degenerated conic sections.
θ1 < θ2 ≤ 90°
θ1 =θ2
0 ≤ θ2 < θ1
A circle is the set of all points in a plane that are equidistant from a fixed point in the plane.
The fixed point is called the centre of the circle.
The fixed distance is called the radius of the circle.
To find the equation of a circle, let us watch the following video
The equation of the circle with radius r and centre (0, 0) is
{x}^{2}+{y}^{2}={r}^{2}
The equation of the circle with centre (a, b) and radius r is
{\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}={r}^{2}
General equation of the circle is
{x}^{2}+{y}^{2}+2gx+2fy+c=0
\left(-g, -f\right)
is the centre and
r=\sqrt{{g}^{2}+{f}^{2}-c}
The equation of a circle with
\mathrm{A}\left({x}_{1}, {y}_{1}\right)
\mathrm{B}\left({x}_{2}, {y}_{2}\right)
\left(x- {x}_{1}\right)\left(x-{x}_{2}\right) +\left(y- {y}_{1}\right)\left(y-{y}_{2}\right)=0
Equation of Circle in Different Conditions
1. The equation of the circle with radius r, touching both the axes and lying in the first quadrant is
{\left(x-r\right)}^{2\dots }
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Limit theorems for U-statistics indexed by a one dimensional random walk
Guillotin-Plantard, Nadine ; Ladret, Véronique
{\left({S}_{n}\right)}_{n\ge 0}
ℤ
-random walk and
{\left({\xi }_{x}\right)}_{x\in ℤ}
be a sequence of independent and identically distributed
ℝ
-valued random variables, independent of the random walk. Let
h
be a measurable, symmetric function defined on
{ℝ}^{2}
ℝ
. We study the weak convergence of the sequence
{𝒰}_{n},n\in ℕ
D\left[0,1\right]
the set of right continuous real-valued functions with left limits, defined by
\phantom{\rule{-56.9055pt}{0ex}}\sum _{i,j=0}^{\left[nt\right]}h\left({\xi }_{{S}_{i}},{\xi }_{{S}_{j}}\right),t\in \left[0,1\right].
Statistical applications are presented, in particular we prove a strong law of large numbers for
U
-statistics indexed by a one-dimensional random walk using a result of [1].
Mots clés : random walk, random scenery,
U
-statistics, functional limit theorem
author = {Guillotin-Plantard, Nadine and Ladret, V\'eronique},
title = {Limit theorems for {U-statistics} indexed by a one dimensional random walk},
AU - Guillotin-Plantard, Nadine
AU - Ladret, Véronique
TI - Limit theorems for U-statistics indexed by a one dimensional random walk
Guillotin-Plantard, Nadine; Ladret, Véronique. Limit theorems for U-statistics indexed by a one dimensional random walk. ESAIM: Probability and Statistics, Tome 9 (2005), pp. 98-115. doi : 10.1051/ps:2005004. http://www.numdam.org/articles/10.1051/ps:2005004/
[1] J. Aaronson, R. Burton, H. Dehling, D. Gilat, T. Hill and B. Weiss, Strong laws for
L
U
-statistics. Trans. Amer. Math. Soc. 348 (1996) 2845-2866. | Zbl 0863.60032
[2] P. Billingsley, Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, second edition. A Wiley-Interscience Publication (1999). | MR 1700749 | Zbl 0944.60003
[3] E. Bolthausen, A central limit theorem for two-dimensional random walks in random sceneries. Ann. Probab. 17 (1989) 108-115. | Zbl 0679.60028
[4] E. Boylan, Local times for a class of Markoff processes. Illinois J. Math. 8 (1964) 19-39. | Zbl 0126.33702
[5] E. Buffet and J.V. Pulé, A model of continuous polymers with random charges. J. Math. Phys. 38 (1997) 5143-5152. | Zbl 0890.60099
[6] P. Cabus and N. Guillotin-Plantard, Functional limit theorems for
U
-statistics indexed by a random walk. Stochastic Process. Appl. 101 (2002) 143-160. | Zbl 1075.60018
[7] F. Den Hollander, Mixing properties for random walk in random scenery. Ann. Probab. 16 (1988) 1788-1802. | Zbl 0651.60108
[8] F. Den Hollander, M.S. Keane, J. Serafin and J.E. Steif, Weak bernoullicity of random walk in random scenery. Japan. J. Math. (N.S.) 29 (2003) 389-406. | Zbl 1049.60041
[9] F. Den Hollander and J.E. Steif, Mixing properties of the generalized
T,{T}^{-1}
-process. J. Anal. Math. 72 (1997) 165-202. | Zbl 0898.60070
[10] R.K. Getoor and H. Kesten, Continuity of local times for Markov processes. Comp. Math. 24 (1972) 277-303. | Numdam | Zbl 0293.60069
[11] W. Hoeffding, The strong law of large numbers for
U
-statistics. Univ. N. Carolina, Institue of Stat. Mimeo series 302 (1961).
[12] H. Kesten and F. Spitzer, A limit theorem related to a new class of self-similar processes. Z. Wahrsch. Verw. Gebiete 50 (1979) 5-25. | Zbl 0396.60037
[13] A.J. Lee,
U
-statistics. Theory and practice. Marcel Dekker, Inc., New York (1990). | MR 1075417 | Zbl 0771.62001
[14] M. Maejima, Limit theorems related to a class of operator-self-similar processes. Nagoya Math. J. 142 (1996) 161-181. | Zbl 0865.60033
[15] S. Martínez and D. Petritis, Thermodynamics of a Brownian bridge polymer model in a random environment. J. Phys. A 29 (1996) 1267-1279. | Zbl 0919.60078
[16] I. Meilijson, Mixing properties of a class of skew-products. Israel J. Math. 19 (1974) 266-270. | Zbl 0305.28008
[17] D. Revuz and M. Yor, Continuous martingales and Brownian motion. Springer-Verlag, Berlin. Fundamental Principles of Mathematical Sciences 293 (1999). | MR 1725357 | Zbl 0917.60006
[18] R.J. Serfling, Approximation theorems of mathematical statistics. John Wiley & Sons Inc., New York. Wiley Series in Probability and Mathematical Statistics (1980). | MR 595165 | Zbl 0538.62002
[19] F. Spitzer, Principles of random walks. Springer-Verlag, New York, second edition. Graduate Texts in Mathematics 34 (1976). | MR 388547 | Zbl 0359.60003
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Molecules | Free Full-Text | 1H NMR Study of the HCa2Nb3O10 Photocatalyst with Different Hydration Levels
Interfacial Compatibilization into PLA/Mg Composites for Improved In Vitro Bioactivity and Stem Cell Adhesion
Study on Nuclear Magnetic Resonance Logging T2 Spectrum Shape Correction of Sandstone Reservoirs in Oil-Based Mud Wells
Modulation of the NOTCH1 Pathway by LUNATIC FRINGE Is Dominant over That of MANIC or RADICAL FRINGE
Silyukov, O. I.
Nefedov, D. Y.
Antonenko, A. O.
Missyul, A.
Kurnosenko, S. A.
Zvereva, I. A.
Oleg I. Silyukov
Elizaveta A. Andronova
Denis Y. Nefedov
Anastasiia O. Antonenko
Alexander Missyul
Sergei A. Kurnosenko
Faculty of Physics, Saint Petersburg State University, 7/9 Universitetskaya nab., 199034 Saint Petersburg, Russia
Institute of Chemistry, Saint Petersburg State University, 7/9 Universitetskaya nab., 199034 Saint Petersburg, Russia
CELLS-ALBA Synchrotron, 08290 Cerdanyola del Vallès, Barcelona, Spain
The photocatalytic activity of layered perovskite-like oxides in water splitting reaction is dependent on the hydration level and species located in the interlayer slab: simple or complex cations as well as hydrogen-bonded or non-hydrogen-bonded H2O. To study proton localization and dynamics in the HCa2Nb3O10·yH2O photocatalyst with different hydration levels (hydrated—α-form, dehydrated—γ-form, and intermediate—β-form), complementary Nuclear Magnetic Resonance (NMR) techniques were applied. 1H Magic Angle Spinning NMR evidences the presence of different proton containing species in the interlayer slab depending on the hydration level. For α-form, HCa2Nb3O10·1.6H2O, 1H MAS NMR spectra reveal H3O+. Its molecular motion parameters were determined from 1H spin-lattice relaxation time in the rotating frame (T1ρ) using the Kohlrausch-Williams-Watts (KWW) correlation function with stretching exponent β = 0.28:
{E}_{\mathrm{a}}=0.210\left(2\right)
{\tau }_{0}=9.0\left(1\right) × {10}^{-12}
s. For the β-form, HCa2Nb3O10·0.8H2O, the only 1H NMR line is the result of an exchange between lattice and non-hydrogen-bonded water protons. T1ρ(1/T) indicates the presence of two characteristic points (224 and 176 K), at which proton dynamics change. The γ-form, HCa2Nb3O10·0.1H2O, contains bulk water and interlayer H+ in regular sites. 1H NMR spectra suggest two inequivalent cation positions. The parameters of the proton motion, found within the KWW model, are as follows:
{E}_{\mathrm{a}}=0.217\left(8\right)
{\tau }_{0}=8.2\left(9\right) × {10}^{-10}
s. View Full-Text
Keywords: layered perovskite-like niobate; Dion-Jacobson phase; proton NMR layered perovskite-like niobate; Dion-Jacobson phase; proton NMR
Shelyapina, M.G.; Silyukov, O.I.; Andronova, E.A.; Nefedov, D.Y.; Antonenko, A.O.; Missyul, A.; Kurnosenko, S.A.; Zvereva, I.A. 1H NMR Study of the HCa2Nb3O10 Photocatalyst with Different Hydration Levels. Molecules 2021, 26, 5943. https://doi.org/10.3390/molecules26195943
Shelyapina MG, Silyukov OI, Andronova EA, Nefedov DY, Antonenko AO, Missyul A, Kurnosenko SA, Zvereva IA. 1H NMR Study of the HCa2Nb3O10 Photocatalyst with Different Hydration Levels. Molecules. 2021; 26(19):5943. https://doi.org/10.3390/molecules26195943
Shelyapina, Marina G., Oleg I. Silyukov, Elizaveta A. Andronova, Denis Y. Nefedov, Anastasiia O. Antonenko, Alexander Missyul, Sergei A. Kurnosenko, and Irina A. Zvereva. 2021. "1H NMR Study of the HCa2Nb3O10 Photocatalyst with Different Hydration Levels" Molecules 26, no. 19: 5943. https://doi.org/10.3390/molecules26195943
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Parallel connected network - MATLAB - MathWorks France
Network of RF Objects In Parallel
Parallel connected network
Use the parallel class to represent networks of linear RF objects connected in parallel that are characterized by the components that make up the network. The following figure shows a pair of networks in a parallel configuration.
h = rfckt.parallel
h = rfckt.parallel('Ckts',value)
h = rfckt.parallel returns a parallel connected network object whose properties all have their default values.
h = rfckt.parallel('Ckts',value) returns a cascaded network with elements specified in the name-value pair property Ckts.
Create a network of transmission lines connected in parallel using rfckt.parallel.
rfplel = rfckt.parallel('Ckts',{tx1,tx2})
rfplel =
rfckt.parallel with properties:
Name: 'Parallel Connected Network'
The analyze method first calculates the admittance matrix of the parallel connected network. It starts by converting each component network's parameters to an admittance matrix. The following figure shows a parallel connected network consisting of two 2-port networks, each represented by its admittance matrix.
\begin{array}{l}\left[{Y}^{\prime }\right]=\left[\begin{array}{cc}{Y}_{11}{}^{\prime }& {Y}_{12}{}^{\prime }\\ {Y}_{21}{}^{\prime }& {Y}_{22}{}^{\prime }\end{array}\right]\\ \left[{Y}^{″}\right]=\left[\begin{array}{cc}{Y}_{11}{}^{\prime \text{}\prime }& {Y}_{12}{}^{\prime \text{}\prime }\\ {Y}_{21}{}^{\prime \text{}\prime }& {Y}_{22}{}^{\prime \text{}\prime }\end{array}\right]\end{array}
The analyze method then calculates the admittance matrix for the parallel network by calculating the sum of the individual admittances. The following equation illustrates the calculations for two 2-port circuits.
\left[Y\right]=\left[{Y}^{\prime }\right]+\left[{Y}^{″}\right]=\left[\begin{array}{cc}{Y}_{11}{}^{\prime }+{Y}_{11}{}^{\prime \text{}\prime }& {Y}_{12}{}^{\prime }+{Y}_{12}{}^{\prime \text{}\prime }\\ {Y}_{21}{}^{\prime }+{Y}_{21}{}^{\prime \text{}\prime }& {Y}_{22}{}^{\prime }+{Y}_{22}{}^{\prime \text{}\prime }\end{array}\right]
Finally, analyze converts the admittance matrix of the parallel network to S-parameters at the frequencies specified in the analyze input argument freq.
rfckt.amplifier | rfckt.cascade | rfckt.coaxial | rfckt.cpw | rfckt.datafile | rfckt.delay | rfckt.hybrid | rfckt.hybridg | rfckt.mixer | rfckt.microstrip | rfckt.passive | rfckt.parallelplate | rfckt.rlcgline | rfckt.series | rfckt.seriesrlc | rfckt.shuntrlc | rfckt.twowire | rfckt.txline
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Critically Damped System (ξ = 1) | Education Lessons
Fig.1(Critically Damped System)
Critically damped system(ξ=1): If the damping factor ξ is equal to one, or the damping coefficient c is equal to critical damping coefficient "cc", then the system is said to be a critically damped system.
\xi=1 \quad \text OR \quad {c \over c_c} = 1\implies c = c_c
Two roots for critically damped system are given by S1 and S2 as below:
S_1 = \big [-\xi + \sqrt{\xi^2 -1} \big] \omega_n \\ S_2 = \big [-\xi - \sqrt{\xi^2 -1} \big] \omega_n
ξ=1
S_1 = S_2 = -\omega_n
Here both the roots are real and equal, so the solution to the differential equation can be given by
x = (A + Bt)e^{-\omega_n t} \quad \quad ...(1)
Now differentiating equation (1) with respect to ‘t’, we get:
\mathring x = Be^{-\omega_nt} - \omega_n(A + Bt)e^{-\omega_nt} \quad \quad ...(2)
Now, let at
t = 0
x = X_0
t = 0
\mathring x =0
Substituting these values in equation (1):
X_0 = A \quad ...(3)
Same way, from equation (2), we get
\begin{aligned} 0&= B - \omega_n(A + 0) \\ 0&= B - \omega_nA \\ B&= \omega_nA \\ B&= \omega_nX_0 \qquad ...(4) \end{aligned}
Now putting the values of A and B in equation (1), we get:
\begin{aligned} x&= (X_0 + \omega_nX_0t)e^{-\omega_nt} \\ x&= X_0(1 + \omega_nt)e^{-\omega_nt} \qquad ...(5) \end{aligned}
From above equation (5), it is seen that as time t increases, the displacement x decreases exponentially.
The motion of a critically damped system is aperiodic (aperiodic motion motions are those motions in which the motion does not repeat after a regular interval of time i.e non periodic motion) and so the system does not shows vibrations.
For critically damped systems, if a system is displaced from its initial position, it will try to reach its mean position in a very short time.
Critically damped systems are generally seen in hydraulic doors closer as it is necessary for the door to come to its initial position in a very short time.
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Calculate fourth-order point mass - Simulink - MathWorks Switzerland
4th Order Point Mass (Longitudinal)
Initial downrange [East]
Initial altitude [Up]
Calculate fourth-order point mass
The 4th Order Point Mass (Longitudinal) block performs the calculations for the translational motion of a single point mass or multiple point masses. For more information on the system for the translational motion of a single point mass or multiple mass, see Algorithms.
The 4th Order Point Mass (Longitudinal) block port labels change based on the input and output units selected from the Units list.
The flat Earth reference frame is considered inertial, an approximation that allows the forces due to the Earth's motion relative to the “fixed stars” to be neglected.
Port_1 — Force in x-axis
Force in x-axis, specified as a scalar or array, in selected units.
Port_2 — Force in z-axis
Force in z-axis, specified as a scalar or array, in selected units.
Port_1 — Flight path angle
Flight path angle, returned as a scalar or array, in radians.
Port_2 — Airspeed
Airspeed, returned as a scalar or array, in selected units.
Port_3 — Downrange or amount traveled east
Downrange or amount traveled east, returned as a scalar or array, in selected units.
Port_4 — Altitude or amount traveled up
Altitude or amount traveled up, returned as a scalar or array, in selected units.
Reference frame orientation — Units
[North East Down] (default) | [East North Down]
Initial flight path angle of the point mass(es), specified as a scalar or vector.
Block Parameter: gamma0
Initial airspeed of the point mass(es), specified as a scalar or vector.
Block Parameter: V0
Initial downrange [East] — Initial downrange
Initial downrange of the point mass(es), specified as a scalar or vector.
Initial altitude [Up] — Initial altitude of point masses
Initial altitude of the point mass(es), specified as a scalar or vector.
Initial mass — Point mass
Mass of the point mass(es), specified as a scalar or vector.
Block Parameter: mass0
The translational motions of the point mass [XEastXUp]T are functions of airspeed (V) and flight path angle (γ),
\begin{array}{c}{F}_{x}=m\stackrel{˙}{V}\\ {F}_{z}=mV\stackrel{˙}{\gamma }\\ {\stackrel{˙}{X}}_{East}=V\mathrm{cos}\gamma \\ {\stackrel{˙}{X}}_{Up}=V\mathrm{sin}\gamma \end{array}
where the applied forces [FxFz]T are in a system defined as follows: x-axis is in the direction of vehicle velocity relative to air, z-axis is upward, and y-axis completes the right-handed frame. The mass of the body m is assumed constant.
Simple Variable Mass 3DOF (Body Axes) | Custom Variable Mass 3DOF (Wind Axes) | 4th Order Point Mass Forces (Longitudinal) | 3DOF (Body Axes) | 3DOF (Wind Axes) | 6th Order Point Mass (Coordinated Flight) | Custom Variable Mass 3DOF (Body Axes) | 6th Order Point Mass Forces (Coordinated Flight) | Simple Variable Mass 3DOF (Wind Axes)
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In Measurement of a Circle, he did this by drawing a larger regular hexagon outside a circle then a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each regular polygon, calculating the length of a side of each polygon at each step. As the number of sides increases, it becomes a more accurate approximation of a circle. After four such steps, when the polygons had 96 sides each, he was able to determine that the value of π lay between 31/7 (approx. 3.1429) and 310/71 (approx. 3.1408), consistent with its actual value of approximately 3.1416.[65] He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle (
{\textstyle \pi r^{2}}
{\displaystyle \sum _{n=0}^{\infty }4^{-n}=1+4^{-1}+4^{-2}+4^{-3}+\cdots ={4 \over 3}.\;}
{\displaystyle \,r=a+b\theta }
with real numbers a and b.
^ Boyer, Carl Benjamin. 1991. A History of Mathematics. ISBN 978-0-471-54397-8: "Arabic scholars inform us that the familiar area formula for a triangle in terms of its three sides, usually known as Heron's formula —
{\displaystyle k={\sqrt {s(s-a)(s-b)(s-c)}}}
{\displaystyle s}
is the semiperimeter — was known to Archimedes several centuries before Heron lived. Arabic scholars also attribute to Archimedes the 'theorem on the broken chord' ... Archimedes is reported by the Arabs to have given several proofs of the theorem."
Retrieved from "https://en.wikipedia.org/w/index.php?title=Archimedes&oldid=1080976639"
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LMIs in Control/Matrix and LMI Properties and Tools/Non-expansivity and Bounded Realness - Wikibooks, open books for an open world
LMIs in Control/Matrix and LMI Properties and Tools/Non-expansivity and Bounded Realness
This section studies the non-expansivity and bounded-realness of a system.
4 LMI Condition
Given a state-space representation of a linear system
{\displaystyle {\begin{aligned}\ {\dot {x}}=Ax+Bu\\\ y=Cx+Du\\\end{aligned}}}
{\displaystyle x\in \mathbb {R} ^{n},y\in \mathbb {R} ^{m},u\in \mathbb {R} ^{r}}
are the state, output and input vectors respectively.
{\displaystyle A,B,C,D}
are system matrices.
The linear system with the same number of input and output variables is called non-expansive if
{\displaystyle {\begin{aligned}\int \limits _{0}^{T}y^{T}y(t)dt\geq \int \limits _{0}^{T}u^{T}u(t)dt\\\end{aligned}}}
hold for any arbitrary
{\displaystyle T\geq 0}
, arbitrary input
{\displaystyle u(t)}
, and the corresponding solution
{\displaystyle y(t)}
of the system with
{\displaystyle x(0)=0}
. In addition, the transfer function matrix
{\displaystyle {\begin{aligned}G(s)&=C(sI-A)^{-1}B+D\\\end{aligned}}}
of system is called is positive real if it is square and satisfies
{\displaystyle {\begin{aligned}\ G^{H}(s)+G(s)\geq I\forall s\in \mathbb {C} ,Re(s)>0\\\end{aligned}}}
LMI ConditionEdit
Let the linear system be controllable. Then, the system is bounded-real if an only if there exists
{\displaystyle P>0}
{\displaystyle {\begin{aligned}\ {\begin{bmatrix}A^{T}P+PA&PB&C^{T}\\B^{T}P&-I&D^{T}\\C&D&-I\end{bmatrix}}<0\\\end{aligned}}}
{\displaystyle {\begin{aligned}\ {\begin{bmatrix}PA+A^{T}P+C^{T}C&PB+C^{T}D\\B^{T}P+D^{T}C&D^{T}D-I\end{bmatrix}}<0\\\end{aligned}}}
This implementation requires Yalmip and Mosek.
https://github.com/ShenoyVaradaraya/LMI--master/blob/main/bounded_realness.m
Thus, it is seen that passivity and positive-realness describe the same property of a linear system, one gives the time-domain feature and the other provides frequency-domain feature of this property.
LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013
Retrieved from "https://en.wikibooks.org/w/index.php?title=LMIs_in_Control/Matrix_and_LMI_Properties_and_Tools/Non-expansivity_and_Bounded_Realness&oldid=3969278"
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Create vector autoregression (VAR) model - MATLAB - MathWorks 日本
AR: {2×2 matrices of NaNs} at lags [1 2 3 ... and 1 more]
\begin{array}{l}{y}_{1,t}=1+0.2{y}_{1,t-1}-0.1{y}_{2,t-1}+0.5{y}_{3,t-1}+1.5t+{\mathrm{ε}}_{1,t}\\ {y}_{2,t}=1-0.4{y}_{1,t-1}+0.5{y}_{2,t-1}+2t+{\mathrm{ε}}_{2,t}\\ {y}_{3,t}=-0.1{y}_{1,t-1}+0.2{y}_{2,t-1}+0.3{y}_{3,t-1}+{\mathrm{ε}}_{3,t}.\end{array}
\mathrm{Σ}=\left[\begin{array}{ccc}0.1& 0.01& 0.3\\ 0.01& 0.5& 0\\ 0.3& 0& 1\end{array}\right].
AR: {2×2 matrices} at lags [1 2 3 ... and 1 more]
{y}_{t}=c+{\mathrm{Φ}}_{1}{y}_{tâ1}+{\mathrm{Φ}}_{2}{y}_{tâ2}+...+{\mathrm{Φ}}_{p}{y}_{tâp}+\mathrm{β}{x}_{t}+\mathrm{δ}t+{\mathrm{ε}}_{t}.
Φj is a numseries-by-numseries matrix of autoregressive coefficients, where j = 1,...,p and Φp is not a matrix containing only zeros.
β is a numseries-by-numpreds matrix of regression coefficients.
δ is a numseries-by-1 vector of linear time-trend values.
εt is a numseries-by-1 vector of random Gaussian innovations, each with a mean of 0 and collectively a numseries-by-numseries covariance matrix Σ. For t ≠s, εt and εs are independent.
\mathrm{Φ}\left(L\right){y}_{t}=c+\mathrm{β}{x}_{t}+\mathrm{δ}t+{\mathrm{ε}}_{t},
\mathrm{Φ}\left(L\right)=Iâ{\mathrm{Φ}}_{1}Lâ{\mathrm{Φ}}_{2}{L}^{2}â...â{\mathrm{Φ}}_{p}{L}^{p}
, Φ(L)yt is the multivariate autoregressive polynomial, and I is the numseries-by-numseries identity matrix.
\begin{array}{l}{y}_{1,t}={c}_{1}+{\mathrm{Ï}}_{11}{y}_{1,tâ1}+{\mathrm{Ï}}_{12}{y}_{2,tâ1}+{\mathrm{β}}_{11}{x}_{1,t}+{\mathrm{β}}_{12}{x}_{2,t}+{\mathrm{β}}_{13}{x}_{3,t}+{\mathrm{δ}}_{1}t+{\mathrm{ε}}_{1,t}\\ {y}_{2,t}={c}_{2}+{\mathrm{Ï}}_{21}{y}_{1,tâ1}+{\mathrm{Ï}}_{22}{y}_{2,tâ1}+{\mathrm{β}}_{21}{x}_{1,t}+{\mathrm{β}}_{22}{x}_{2,t}+{\mathrm{β}}_{23}{x}_{3,t}+{\mathrm{δ}}_{2}t+{\mathrm{ε}}_{2,t}.\end{array}
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Bending Capacity Analyses of Corroded Pipeline | J. Offshore Mech. Arct. Eng. | ASME Digital Collection
Weiwei Yu,
e-mail: weiwei.yu@chevron.com
Pedro M. Vargas,
Pedro M. Vargas
e-mail: pedrovargas@chevron.com
Dale G. Karr
e-mail: dgkarr@umich.edu
Yu, W., Vargas, P. M., and Karr, D. G. (December 5, 2011). "Bending Capacity Analyses of Corroded Pipeline." ASME. J. Offshore Mech. Arct. Eng. May 2012; 134(2): 021701. https://doi.org/10.1115/1.4004521
Appendix G of the ASME B31 pipeline and piping codes addresses the pressure containment capacity of pipelines and vessels with locally corroded sections. However, the ability of corroded sections to carry moment, for example, in thermal loops, is not addressed in fitness-for-service codes today. This paper presents nonlinear Finite Element Analysis (FEA) and full-scale 4-point-bend testing of pipes with locally-thinned-areas (LTAs) to simulate corrosion. The LTAs are loaded in compression, and the buckle moment is used as the carrying capacity of the corroded section. The nonlinear FEA is found to match the experimental results, validating this methodology for computing moment capacity in corroded sections. Significant secondary effects were found to affect the testing results. This paper identifies and quantifies these effects. Also, somewhat contrary to intuition, internal pressure is demonstrated to adversely affect the bending capacity for the intermediate-low
D/t
ratio (17.25) pipe tested.
bending strength, corrosion, finite element analysis, pipelines
Corrosion, Finite element analysis, Pipelines, Pipes, Pressure, Testing, Tension, Stress, Compression
ASME, 1991, “
,” ASME Tech. Report No. B31G.
Fracture and General Yielding for Carbon Steel Pipes With Local Wall Thinning
ASME J. Pressure Vessel Technology
Limit Moment of Local Wall Thinning in Pipe Under Bending
Modified Expression for Estimating the Limit Bending Moment of Local Corroded Pipeline
Y. -J
Reference Stress Based Approach to Predict Failure Strength of Pipes With Local Wall Thinning Under Single Loading
Finite Element Based Plastic Limit Loads for Cylinders With Part-Through Surface Cracks Under Combined Loading
ABAQUS/Standard Users Manual, Version 6.7, Hibbitt, Karlsson and Sorensen, Inc., Pawtucket, RI.
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Structural similarity (SSIM) index for measuring image quality - MATLAB ssim - MathWorks France
Calculate Structural Similarity Index (SSIM)
Calculate SSIM for dlarray Input
RegularizationConstants
ssimval
ssimmap
Structural similarity (SSIM) index for measuring image quality
ssimval = ssim(A,ref,Name,Value)
[ssimval,ssimmap] = ssim(___)
ssimval = ssim(A,ref) calculates the structural similarity (SSIM) index for grayscale image or volume A using ref as the reference image or volume. A value closer to 1 indicates better image quality.
ssimval = ssim(A,ref,Name,Value) calculates the SSIM, using name-value pairs to control aspects of the computation.
[ssimval,ssimmap] = ssim(___) also returns the local SSIM value for each pixel or voxel in A.
Read an image into the workspace. Create another version of the image, applying a blurring filter.
ref = imread("pout.tif");
H = fspecial("Gaussian",[11 11],1.5);
A = imfilter(ref,H,"replicate");
Display both images as a montage. The images differ most along sharp high-contrast regions, such as the edges of the trellis.
montage({ref,A})
title("Reference Image (Left) vs. Blurred Image (Right)")
Calculate the global SSIM value for the image and local SSIM values for each pixel.
[ssimval,ssimmap] = ssim(A,ref);
Display the local SSIM map. Include the global SSIM value in the figure title. Small values of local SSIM appear as dark pixels in the local SSIM map. Regions with small local SSIM value correspond to areas where the blurred image noticeably differs from the reference image. Large values of local SSIM value appear as bright pixels. Regions with large local SSIM correspond to uniform regions of the reference image, where blurring has less of an impact on the image.
imshow(ssimmap,[])
title("Local SSIM Map with Global SSIM Value: "+num2str(ssimval))
A = imgaussfilt(ref,1.5,"FilterSize",11,"Padding","replicate");
Display both images as a montage.
montage({ref A})
Simulate batches of images by replicating the reference image and the blurred image 16 times along the 4th dimension.
A = repmat(A,[1 1 1 16]);
ref = repmat(ref,[1 1 1 16]);
Create formatted dlarray objects for the reference image batch and the blurred image batch. The format is "SSCB", for spatial-spatial-channel-batch.
A = dlarray(single(A),"SSCB");
ref = dlarray(single(ref),"SSCB");
Calculate the global SSIM value for the image and local SSIM values for each pixel. ssimVal returns a scalar SSIM value for each image in the batch. ssimMap returns a map of SSIM values, the same size as the image, for each image in the batch.
size(ssimVal)
size(ssimMap)
291 240 1 16
A — Image for quality measurement
Image for quality measurement, specified as a numeric array or a dlarray (Deep Learning Toolbox) object. If A is not a 2-D grayscale image or 3-D grayscale volume, such as an RGB image or stack of grayscale images, specify the DataFormat name-value argument. Do not specify the DataFormat name-value argument if A is a formatted dlarray object.
Reference image against which to measure quality, specified as a numeric array or a dlarray (Deep Learning Toolbox) object of the same size and data type as A. If ref is not a 2-D grayscale image or 3-D grayscale volume, such as an RGB image or stack of grayscale images, specify the DataFormat name-value argument. Do not specify the DataFormat name-value argument if ref is a formatted dlarray object.
Example: ssim(A,ref,"DynamicRange",100)
The format cannot include more than one channel label or batch label. Do not specify the DataFormat name-value argument when the input images are formatted dlarray objects.
Example: "SSC" indicates that the array has two spatial dimensions and one channel dimension, appropriate for 2-D RGB image data.
Example: "SSCB" indicates that the array has two spatial dimensions, one channel dimension, and one batch dimension, appropriate for a sequence of 2-D RGB image data.
DynamicRange — Dynamic range of the input image
diff(getrangefromclass(A)) (default) | positive scalar
Dynamic range of the input image, specified as a positive scalar. The default value of "DynamicRange" depends on the data type of image A, and is calculated as diff(getrangefromclass(A)). For example, the default dynamic range is 255 for images of data type uint8, and the default is 1 for images of data type double or single with pixel values in the range [0, 1].
Exponents — Exponents for luminance, contrast, and structural terms
[1 1 1] (default) | 3-element vector of nonnegative numbers
Exponents for the luminance, contrast, and structural terms, specified as a 3-element vector of nonnegative numbers of the form [alpha beta gamma].
Radius — Standard deviation of isotropic Gaussian function
Standard deviation of isotropic Gaussian function, specified as a positive number. This value is used for weighting the neighborhood pixels around a pixel for estimating local statistics. This weighting is used to avoid blocking artifacts in estimating local statistics.
RegularizationConstants — Regularization constants for luminance, contrast, and structural terms
3-element vector of nonnegative numbers
Regularization constants for the luminance, contrast, and structural terms, specified as a 3-element vector of nonnegative numbers of the form [c1 c2 c3]. The ssim function uses these regularization constants to avoid instability for image regions where the local mean or standard deviation is close to zero. Therefore, small non-zero values should be used for these constants.
C1 = (0.01*L).^2, where L is the specified DynamicRange value.
C3 = C2/2
ssimval — SSIM index
SSIM index, returned as one of these values.
Formatted numeric arrays with neither a channel ("C") nor batch ("B") dimension
Numeric scalar with a single SSIM measurement.
Scalar dlarray object with a single SSIM measurement.
Numeric arrays with a channel or batch dimension specified using the DataFormat name-value argument
Numeric array of the same dimensionality as the input images. The spatial dimensions of ssimval are singleton dimensions. There is one SSIM measurement for each element along any channel or batch dimension.
Formatted dlarray objects with a channel or batch dimension
Unformatted dlarray objects with a channel or batch dimension specified using the DataFormat name-value argument
dlarray object of the same dimensionality as the input images. The spatial dimensions of ssimval are singleton dimensions. There is one SSIM measurement for each element along any channel or batch dimension.
ssimval is of data type double except when A is of data type single, in which case ssimval is of data type single.
The value of ssimval is typically in the range [0, 1]. The value 1 indicates the highest quality and occurs when A and ref are equivalent. Smaller values correspond to poorer quality. For some combinations of inputs and name-value pair arguments, ssimval can be negative.
ssimmap — Local values of SSIM index
Local values of the SSIM index, returned as one of these values.
Numeric array the same size as the input images. There is one SSIM measurement for each element in the input image.
dlarray object the same size as the input images. There is one SSIM measurement for each element in the input image.
Numeric array the same size as the input images. Each spatial element in the input image has an SSIM measurement along any channel or batch dimension.
dlarray object the same size as the input images. Each spatial element in the input image has an SSIM measurement along any channel or batch dimension.
ssimmap is of data type double except when A is of data type single, in which case ssimmap is of data type single.
An image quality metric that assesses the visual impact of three characteristics of an image: luminance, contrast and structure.
If A and ref specify RGB image data, use the "DataFormat" name-value argument to label the channel dimension, "C". You can then apply the mean function along the channel dimension of ssimval and ssimmap to approximate the SSIM index for the overall image.
The SSIM Index quality assessment index is based on the computation of three terms, namely the luminance term, the contrast term and the structural term. The overall index is a multiplicative combination of the three terms.
SSIM\left(x,y\right)={\left[l\left(x,y\right)\right]}^{\alpha }\cdot {\left[c\left(x,y\right)\right]}^{\beta }\cdot {\left[s\left(x,y\right)\right]}^{\gamma }
\begin{array}{l}l\left(x,y\right)=\frac{2{\mu }_{x}{\mu }_{y}+{C}_{1}}{{\mu }_{x}^{2}+{\mu }_{y}^{2}+{C}_{1}},\\ c\left(x,y\right)=\frac{2{\sigma }_{x}{\sigma }_{y}+{C}_{2}}{{\sigma }_{x}^{2}+{\sigma }_{y}^{2}+{C}_{2}},\\ s\left(x,y\right)=\frac{{\sigma }_{xy}+{C}_{3}}{{\sigma }_{x}{\sigma }_{y}+{C}_{3}}\end{array}
where μx, μy, σx,σy, and σxy are the local means, standard deviations, and cross-covariance for images x, y. If α = β = γ = 1 (the default for Exponents), and C3 = C2/2 (default selection of C3) the index simplifies to:
SSIM\left(x,y\right)=\frac{\left(2{\mu }_{x}{\mu }_{y}+{C}_{1}\right)\left(2{\sigma }_{xy}+{C}_{2}\right)}{\left({\mu }_{x}^{2}+{\mu }_{y}^{2}+{C}_{1}\right)\left({\sigma }_{x}^{2}+{\sigma }_{y}^{2}+{C}_{2}\right)}
When you specify a noninteger value for "Exponents", the ssim function prevents complex valued outputs by clamping the intermediate luminance, contrast, and structural terms to the range [0, inf].
[1] Zhou, W., A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli. "Image Quality Assessment: From Error Visibility to Structural Similarity." IEEE Transactions on Image Processing. Vol. 13, Issue 4, April 2004, pp. 600–612.
psnr | immse | multissim | multissim3
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Revision as of 04:54, 14 August 2016 by NikosA (talk | contribs) (i.fusion.hpf)
{\displaystyle {\frac {W}{m^{2}*sr*nm}}}
{\displaystyle L\lambda ={\frac {10^{4}*DN\lambda }{CalCoef\lambda *Bandwidth\lambda }}}
{\displaystyle \rho _{p}={\frac {\pi *L\lambda *d^{2}}{ESUN\lambda *cos(\Theta _{S})}}}
{\displaystyle \rho }
{\displaystyle \pi }
{\displaystyle L\lambda }
{\displaystyle d}
{\displaystyle Esun}
{\displaystyle cos(\theta _{s})}
{\displaystyle {\frac {W}{m^{2}*\mu m}}}
Pan-Sharpening / Fusion is the process of merging high-resolution panchromatic and lower resolution multi-spectral imagery. GRASS 7 holds a dedicated pan-sharpening module, i.pansharpen which features three techniques for sharpening, namely the Brovey transformation, the classical IHS method and one that is based on Principal Components Analysis (PCA). Another algorithm deriving excellent detail and a realistic representation of original multispectral scene colors, is the High-Pass Filter Addition (HPFA) technique. It is available through the add-on i.fusion.hpf (src) (for GRASS 6, please refer to a bash shell script https://github.com/NikosAlexandris/i.fusion.hpf which is, however, unmaintained).
{\displaystyle [0,255]}
{\displaystyle [0,2047]}
The process involves a convolution using a High Pass Filter (HPF) on the high resolution data, then combining this with the lower resolution multispectral data.
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Straight Lines, Popular Questions: CBSE Class 11-commerce SCIENCE, Science - Meritnation
2{x}^{2}-b{y}^{2}+\left(2b-1\right)xy-x-by=0
b+{b}^{2}
the base of an equilateral triangle with side 2a lies along y-axis such that the midpoint of the base is at the origin.find the vertices of the triangle.
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ARIMA Model Specifications - MATLAB & Simulink - MathWorks 日本
ARIMA Model with Known Parameter Values
Specify ARIMA Model Using Econometric Modeler App
This example shows how to use the shorthand arima(p,D,q) syntax to specify the default ARIMA(p, D, q) model,
{\mathrm{Î}}^{D}{y}_{t}=c+{\mathrm{Ï}}_{1}{\mathrm{Î}}^{D}{y}_{t-1}+â¦+{\mathrm{Ï}}_{p}{\mathrm{Î}}^{D}{y}_{t-p}+{\mathrm{ε}}_{t}+{\mathrm{θ}}_{1}{\mathrm{ε}}_{t-1}+â¦+{\mathrm{θ}}_{q}{\mathrm{ε}}_{t-q},
{\mathrm{Î}}^{D}{y}_{t}
{D}^{th}
differenced time series. You can write this model in condensed form using lag operator notation:
\mathrm{Ï}\left(L\right)\left(1-L{\right)}^{D}{y}_{t}=c+\mathrm{θ}\left(L\right){\mathrm{ε}}_{t}.
By default, all parameters in the created model object have unknown values, and the innovation distribution is Gaussian with constant variance.
Specify the default ARIMA(1,1,1) model:
The output shows that the created model object, Mdl, has NaN values for all model parameters: the constant term, the AR and MA coefficients, and the variance. You can modify the created model using dot notation, or input it (along with data) to estimate.
The property P has value 2 (p + D). This is the number of presample observations needed to initialize the AR model.
This example shows how to specify an ARIMA(p, D, q) model with known parameter values. You can use such a fully specified model as an input to simulate or forecast.
Specify the ARIMA(2,1,1) model
\mathrm{Î}{y}_{t}=0.4+0.8\mathrm{Î}{y}_{t-1}-0.3\mathrm{Î}{y}_{t-2}+{\mathrm{ε}}_{t}+0.5{\mathrm{ε}}_{t-1},
where the innovation distribution is Student's t with 10 degrees of freedom, and constant variance 0.15.
Mdl = arima('Constant',0.4,'AR',{0.8,-0.3},'MA',0.5,...
'D',1,'Distribution',tdist,'Variance',0.15)
The name-value pair argument D specifies the degree of nonseasonal integration (D).
In the Econometric Modeler app, you can specify the lag structure, presence of a constant, and innovation distribution of an ARIMA(p,D,q) model by following these steps. All specified coefficients are unknown but estimable parameters.
On the Econometric Modeler tab, in the Models section, click ARIMA. To create ARIMAX models, see ARIMAX Model Specifications.
The ARIMA Model Parameters dialog box appears.
Specify the lag structure. To specify an ARIMA(p,D,q) model that includes all AR lags from 1 through p and all MA lags from 1 through q, use the Lag Order tab. For the flexibility to specify the inclusion of particular lags, use the Lag Vector tab. For more details, see Specifying Univariate Lag Operator Polynomials Interactively. Regardless of the tab you use, you can verify the model form by inspecting the equation in the Model Equation section.
To specify an ARIMA(3,1,2) model that includes a constant, includes all consecutive AR and MA lags from 1 through their respective orders, and has a Gaussian innovation distribution:
To specify an ARIMA(3,1,2) model that includes all AR and MA lags from 1 through their respective orders, has a Gaussian distribution, but does not include a constant:
To specify an ARIMA(8,1,4) model containing nonconsecutive lags
\left(1â{\mathrm{Ï}}_{1}Lâ{\mathrm{Ï}}_{4}{L}^{4}â{\mathrm{Ï}}_{8}{L}^{8}\right)\left(1âL\right){y}_{t}=\left(1+{\mathrm{θ}}_{1}{L}^{1}+{\mathrm{θ}}_{4}{L}^{4}\right){\mathrm{ε}}_{t},
Click the Lag Vector tab.
To specify an ARIMA(3,1,2) model that includes all consecutive AR and MA lags through their respective orders and a constant term, and has t-distribution innovations:
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Statistics of Linear Regression Practice Problems Online | Brilliant
If you’ve ever taken a class in statistics before, linear regression is probably a familiar concept. This is no coincidence. At its core, machine learning is about taking in information and expanding on it, so it's natural that techniques from statistics play an important role in machine learning.
It is possible to use statistical techniques to find a best-fit line, by first calculating five values about our data. If we represent our data sets as collections of points on a scatter plot, these values are the means of
x
y
, the standard deviations of
x
y
, and the correlation coefficient.
n
data points, then the mean of
x
is simply the sum of all
x
values divided by
n
. Correspondingly, the mean of
y
y
n
After calculating the means
(
typically denoted as
\overline{x}
\overline{y}),
we can find the (sample) standard deviations for the data set through the following formulae:
\begin{aligned} SD_x &= \sqrt{\frac{1}{(n-1)} \cdot \sum_{i=1}^{n} (x_i - \overline{x})^2} \\\\ SD_y &= \sqrt{\frac{1}{(n-1)} \cdot \sum_{i=1}^{n} (y_i - \overline{y})^2}. \end{aligned}
The standard deviation of a data set gives a good idea of how close an average data point will be to the mean. A low standard deviation means that data points tend to cluster around the mean, while a large standard deviation usually means that they will be more spread out.
Out of the following sets of numbers, which will have the highest standard deviation? Use your intuition!
[1, 4, 3, 5, 1]
[50, 52, 48, 48, 50]
[90, 90, 90, 90, 90]
[15, 40, 10, 18, 31]
Say that we are studying the correlation between voltage and a light bulb's brightness. Amazingly, we always know exactly what voltage we are using, but we haven’t been so lucky when measuring bulb brightness. Here are our options:
A perfectly accurate sensor taped to another light bulb, twice as bright as the one we’re measuring. We can’t get rid of the second light bulb; we don’t know why but the sensor won’t work without it.
A primitive but effective device from the 1840’s, that looks like a thermometer. It works, but a human must estimate its readings.
A completely broken machine. It always just reads zero.
Actual state-of-the-art technology, no quirks attached.
If we always run the light bulb with the same voltage and measure the brightness ten times, which of these devices will collect data with the highest standard deviation, all else being equal? Assume that the lightbulb is perfectly consistent and not a source of any fluctuations.
After we have
SD_x
SD_y,
we only have the correlation coefficient, usually denoted by
r,
left to calculate. This is difficult to calculate by hand, but the process for doing so is actually quite simple.
The first step is to convert
x
y
to standard units. For
1 \leq i \leq n,
we must put each value
x_i
\frac{x_i-\overline{x}}{SD_x},
which outputs the number of standard deviations
x_i
is above the mean. We will call the updated value for
x_i
x_i^*
. Each value
y_i
should be put through the analogous process with
\overline{y}
SD_y.
Now that we’ve changed our units, we can find our correlation coefficient by taking the average of the product of the updated
x
y
for each of our points.
r = \frac{1}{n} \cdot \sum_{i=1}^{n} (x_i^*\cdot y_i^*).
The formula for the correlation coefficient may seem inscrutable, but it’s actually quite easy to interpret. Its value ranges from
-1
1,
and it indicates how linearly correlated the data is.
r
is close to zero, then the data is barely correlated at all, at least not with a linear relationship. However, if
r
1,
then the data is correlated and can be approximated well by a best-fit line with a positive slope. Conversely, if
r
-1,
the data is correlated and can be approximated well by a best-fit line with a negative slope.
Which of the following values would be closest to the correlation coefficient of the graph below?
-1 3 0.25 1
Now, with all this information, we can finally calculate a best-fit line. In standard units, this is simply the line
y^* = rx^*,
r
is the correlation coefficient. A little algebra will show that in the original units, this translates to the equation
y - \overline{y} = \frac{rSD_y}{SD_x}(x-\overline{x}).
In other words, this version of the best-fit line has a slope of
\frac{rSD_y}{SD_x}
and must pass through the point
(\overline{x}, \overline{y})
Although this line cannot go through every point in the data set, it does do a good job of representing them as a whole. It usually gives a good estimate for the expected value of
y
x,
at least if the relationship between the two is somewhat linear.
Alfred has worked hard analyzing his data, but he needs help with the last step. He has calculated that
\overline{x} = 50,
\overline{y} = 30,
SD_x=8,
SD_y=16,
r=0.75.
Which of the following equations gives the best-fit line he needs?
y = 1.5 x
y - 30 = 1.5 (x-50)
y = 0.375 x + 48.75
y = 0.66 x
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Enhanced downlink control information search - MATLAB lteEPDCCHSearch - MathWorks India
lteEPDCCHSearch
Search EPDCCH for DCI Messages
EPDCCH Search Processing
Enhanced downlink control information search
[dcistr,dcibits] = lteEPDCCHSearch(enb,chs,softbits)
[dcistr,dcibits] = lteEPDCCHSearch(enb,chs,softbits) recovers DCI message structures, and corresponding vectors of DCI message bits, after blind decoding the multiplexed EPDCCHs. The multiplexed EPDCCHs are within the received EPDCCH payload given by matrix of soft bits. This function carries out search for a single EPDCCH set. For more information, see EPDCCH Search Processing.
Encode a DCI message and modulate it on the EPDCCH. Perform EPDCCH decoding and then EPDCCH blind-search to recover the DCI message. For DCI messages sent on the EPDCCH, set the ControlChannelType to 'EPDCCH'.
Initialize cell-wide settings structure and EPDCCH transmission channel structure.
chs.EPDCCHPRBSet = [2 3];
Create a DCI message. Generate EPDCCH candidates.
[dci,dciBits] = lteDCI(enb,chs,struct('DCIFormat',chs.DCIFormat));
Generate RE grid indices and EPDCCH info structure. Encode the DCI message into a codeword for transmission. Generate EPDCCH symbols and populate resource grid.
Decode the EPDCCH transmission. Recover and view DCI message.
rxsoftbits = lteEPDCCHDecode(enb,chs,grid);
rxdci = lteEPDCCHSearch(enb,chs,rxsoftbits);
rxdci{1}
Search for multiple EPDCCH sets. The first EPDCCH set is as configured above and the second is of Distributed type with 8 PRBs.
Transmit the EPDCCH DM-RS for channel estimation.
Configure the channel estimation.
Configure two EPDCCH sets.
chs.EPDCCHTypeList = {'Localized' 'Distributed'};
chs.EPDCCHPRBSetList = {[2; 3] (8:15).'};
Perform the EPDCCH search for each set.
for p = 1:numel(chs.EPDCCHTypeList)
chs.EPDCCHType = chs.EPDCCHTypeList{p};
chs.EPDCCHPRBSet = chs.EPDCCHPRBSetList{p};
rxsoftbits = lteEPDCCHDecode(enb,chs,grid,hestgrid,noiseest);
X = ['EPDCCH set ' num2str(p)];
disp([X ', DCI messages found: ' num2str(numel(rxdci))])
if (~isempty(rxdci))
EPDCCH set 1, DCI messages found: 1
A DCI message is found in EPDCCH set 1 but not in EPDCCH set 2.
{N}_{\text{RB}}^{\text{DL}}
See Specifying Number of Resource Blocks.
{N}_{\text{RB}}^{\text{UL}}
The following parameter is applicable only when chs.EPDCCHStart is absent.
The following zero power CSI-RS resource parameter is applicable only if one or more of the above zero power subframe configurations are set to any value other than 'Off'.
EPDCCH nID parameter for scrambling sequence initialization.
EPDCCHPRBSetList Optional
PRB pair indices for one or two EPDCCH sets.
EPDCCHPRBTypeList Optional
cell array of character vector or string array
EPDCCH transmission types for one or two EPDCCH sets.
EnableCarrierIndication Optional
UE configured with carrier indication field (affects presence of CIF)
EnableSRSRequest Optional
UE configured for SRS request (affects presence of SRS request field in UE-specific formats 0/1A and 2B/2C/2D TDD)
EnableMultipleCSIRequest Optional
UE configured for multiple CSI requests (multiple cells/CSI processes) (affects length of CSI request field in UE-specific formats 0/4)
Number of UE transmission antennas (affects length of precoding information field in format 4)
softbits — Received EPDCCH payload
MTot-by-4 matrix
Received EPDCCH payload containing coded Downlink Control Information (DCI), specified as a MTot-by-4 matrix. MTot is the total number of bits associated with EPDCCHs,
\frac{nEPDCCH*NECCE}{NECCEPerPRB*2}
. The matrix contains soft EPDCCH bits estimates for all EPDCCH ECCEs and all EPDCCH reference signal ports.
If chs.EPDCCHPRBSetList and chs.EPDCCHTypeList are present and each contain two elements, the creation of the EPDCCH candidate locations support two EPDCCH sets. For more information, see TS 36.213 [2], Tables 9.1.4-3a to 9.1.4-5b.
DCI message structure, returned as a cell array of structures whose fields match of the fields associated DCI format.
The field names associated with dcistr depend on the DCI format. The format is expected to be one of the formats generated by lteDCI.
DCISTRFields
'Format0' DCIFormat — 'Format0'
FreqHopping 1-bit PUSCH frequency hopping flag
Allocation variable Resource block assignment/allocation
ModCoding 5-bits Modulation, coding scheme, and redundancy version
NewData 1-bit New data indicator
TPC 2-bits PUSCH TPC command
CShiftDMRS 3-bits Cyclic shift for DM RS
CQIReq 1-bit CQI request
TDDIndex 2-bits
For TDD Config 1-6, this field is the Downlink Assignment Index.
AllocationType 1-bit Resource allocation header: type 0, type 1 (only if downlink bandwidth is >10 PRBs)
ModCoding 5-bits Modulation and coding scheme
3-bits (FDD)
4-bits (TDD)
RV 2-bits Redundancy version
TPCPUCCH 2-bits PUCCH TPC command
TDDIndex 2-bits For TDD config 0, this field is not used. For TDD Config 1-6, this field is the Downlink Assignment Index. Not present for FDD.
'Format1A' DCIFormat — 'Format1A'
AllocationType 1-bit VRB assignment flag: 0 (localized), 1 (distributed)
'Format1B' DCIFormat — 'Format1B'
2-bits (2 antennas)
PMI 1-bit PMI confirmation
'Format1C' DCIFormat — 'Format1C'
'Format1D' DCIFormat — 'Format1D'
DlPowerOffset 1-bit Downlink power offset
SwapFlag 1-bit Transport block to codeword swap flag
ModCoding1 5-bits Modulation and coding scheme for transport block 1
NewData1 1-bit New data indicator for transport block 1
RV1 2-bits Redundancy version for transport block 1
3-bits (2-antennas)
ScramblingId 1-bit Scrambling identity
CIF variable Carrier indicator
TxIndication 3-bits Antenna port(s), scrambling identity, and number of layers indicator
SRSRequest variable SRS request. Only present for TDD.
TPCCommands variable TPC commands for PUCCH and PUSCH
CShiftDMRS 3-bits Cyclic shift for DMRS
TDDIndex 2-bits For TDD config 0, this field is Uplink Index. For TDD Config 1-6, this field is the Downlink Assignment Index. Not present for FDD.
CQIReq variable CQI request
SRSRequest 2-bits SRS request
AllocationType 1-bit Resource allocation header: non-hopping PUSCH resource allocation type 0, type 1
ModCoding 5-bits Modulation, coding scheme and redundancy version
EPDCCH search processing blindly decodes DCI messages based on their lengths. The lengths and order in which the DCI messages are searched for is provided by lteDCIInfo. For DCI messages conveyed on the EPDCCH, set ControlChannelType to 'EPDCCH' when calling lteDCIInfo.
If one or more messages have the same length, the first message format in the list is used to decode the message. The other potential message formats are ignored. The lteEPDCCHSearch function does not consider transmission mode during blind search, and no DCI message format is filtered based on transmission mode. It does not search for format 3 and 3A (power adjustment commands for PUSCH and PUCCH). It also does not search for format 1C as this format is never used in the UE-specific search space. EPDCCH is never used for common search space messages. For more information on the association between transmission mode, transmission scheme, DCI format, and search space, see TS 36.213 [2], Section 7.1 and Table 7.1-5A.
lteEPDCCH | lteEPDCCHDecode | lteEPDCCHIndices | lteEPDCCHSpace | lteEPDCCHPRBS
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Riemann Sums and Definite Integrals Practice Problems Online | Brilliant
\displaystyle{\int_0^{2} 7x^{2}dx}
using a right Riemann sum by dividing the interval into
4
\frac{1683}{64}
\frac{841}{32}
\frac{1681}{64}
\frac{105}{4}
\displaystyle{\int_{0}^{4}x^{5}dx}
using a left Riemann sum?
\displaystyle{\sum_{k=1}^{{4}n}\left(\frac{4}{n}k\right)^{5}\frac{4}{n}}
\displaystyle{\sum_{k=0}^{n-1}\left(\frac{4}{n}k\right)^{5}\frac{4}{n}}
\displaystyle{\sum_{k=1}^{n}\left(\frac{4}{n}k\right)^{5}\frac{4}{n}}
\displaystyle{\sum_{k=0}^{{4}n-1}\left(\frac{4}{n}k\right)^{5}\frac{4}{n}}
\displaystyle{\int_0^1 (6x^2+2)dx}
using a right Riemann sum, by dividing the interval into
7
\frac{218}{49}
\frac{216}{49}
\frac{215}{49}
\frac{31}{7}
The following is Alex's approximation of an integration by using a right Riemann sum:
\frac{9}{5}\cdot\left(\left(\frac{3}{5}\right)^{7}+\left(\frac{6}{5}\right)^{7}+\left(\frac{9}{5}\right)^{7}+\left(\frac{12}{5}\right)^{7}+\left(\frac{15}{5}\right)^{7}\right).
Which of the following integrals is Alex approximating?
\displaystyle{\int_0^{7}{3}x^{5}dx}
\displaystyle{\int_0^{3}{3}x^{7}dx}
\displaystyle{\int_0^{3}{3}x^{8}dx}
\displaystyle{\int_0^{7}{3}x^{4}dx}
What is the Riemann sum of the function
f(x)= x^3-6x
[0, 6]
, if we divide it into 3 equal parts and use the midpoint of each interval?
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Calculus/Pressure and force - Wikibooks, open books for an open world
Calculus/Pressure and force
2 Derivative definition of Force(Instantaneous Force)
3 Effects of Force
Force is a fundamental concept in physics. It is defined as a push or pull on a point mass. The magnitude of this physical quantity is given algebraically as a linear equation by the newton's second law of motion.
{\displaystyle \ F=\ ma}
{\displaystyle \ m}
is the mass of the point particle in kilograms and
{\displaystyle \ a}
is the rate of change of velocity with respect to time (acceleration) in ms-2 (the force obtained will be in newtons).
Derivative definition of Force(Instantaneous Force)Edit
Force (F) is also defined as the rate of change of linear momentum (p) with respect to time (t):
{\displaystyle \ F={\frac {dp}{dt}}}
Effects of ForceEdit
Force plays an important role in the motion of objects as it can cause these effects on the object:
It can make an object at rest acquire motion.
It can make an object in motion come to rest.
It can change the state of motion of an object.
It can change the shape and size of an object.
It can change physical and chemical properties of an object.
It can change the direction of motion of an object.
Pressure (P) is defined as force acting normally on an object per unit area of its surface. It always acts parallel to the surface of an object.
{\displaystyle \ P={\frac {F}{A}}}
{\displaystyle \ F}
is the force acting in newtons and
{\displaystyle \ A}
is the area on which the force is acting on the body in m2 (the pressure obtained will be in pascals).
{\displaystyle \ P={\frac {dF}{dA}}}
gives the instantaneous pressure.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Calculus/Pressure_and_force&oldid=3585882"
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Hyperbolic Geometry | Brilliant Math & Science Wiki
Patrick Corn, Zeno Rogue, Jimin Khim, and
Hyperbolic geometry is a type of non-Euclidean geometry that arose historically when mathematicians tried to simplify the axioms of Euclidean geometry, and instead discovered unexpectedly that changing one of the axioms to its negation actually produced a consistent theory. Later, physicists discovered practical applications of these ideas to the theory of special relativity.
Hyperbolic geometry also inspired the art of M. C. Escher, and has various theoretical applications as well, including geometric group theory and the theory of modular forms.
Results in Hyperbolic Geometry
The first four axioms of Euclidean geometry, laid out in Euclid's Elements, are essentially self-evident:
(1) Any two points can be connected by a line.
(2) Any line segment can be extended indefinitely.
(3) Given a line segment, a circle can be drawn with center at one of the endpoints and radius equal to the length of the segment.
(4) Any two right angles are congruent.
These four axioms define what is sometimes called absolute geometry. Euclid's first 28 propositions used only these four axioms, but he was forced to add a fifth axiom which was much less obvious than the first four:
(5) The parallel postulate: If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
This statement is a bit complicated and wordy; many modern accounts of Euclidean geometry use a different fifth axiom known as Playfair's axiom:
(5') Playfair's axiom: Given a line
L
P
not on
L
, there is at most one line that can be drawn through
P
L
(
In fact, axioms (1)-(4) imply that such a line always exists, by dropping a perpendicular to
L
P
and then dropping a perpendicular to the perpendicular at
P.)
It can be shown that (1), (2), (3), (4), (5)
\Leftrightarrow
(1), (2), (3), (4), (5') logically.
Many mathematicians, going back to the ancient Greeks, attempted to show that (5)
(
or (5')
were actually consequences of (1)-(4). One method that many of them used was proof by contradiction; start with the axioms (1), (2), (3), (4), and the negation of (5), and try to produce something that is "wrong." A lot of progress was made along these lines, and the properties of the subsequent geometry that was produced were quite strange and different from the propositions of Euclid's geometry. But there was nothing logically inconsistent about them. Eventually, 19
^\text{th}
-century mathematicians (starting with the Russian mathematician Nikolai Ivanovich Lobachevsky) shifted from trying to prove that non-Euclidean geometries were impossible, and began instead to explore the consequences of changing the fifth axiom.
In fact, the fifth axiom cannot be proved (or disproved) from the first four, a fact that was established in the late
19^\text{th}
century by the Italian mathematician Eugenio Beltrami and others.
Consider the geometry obtained from replacing Playfair's axiom by its negation:
Given a line
L
P
L
, there are at least two distinct lines that can be drawn through
P
that are parallel to
L
Here are some consequences of these axioms:
(1) The interior angles of a triangle sum to less than
180^\circ.
(2) The interior angles of a quadrilateral sum to less than
360^\circ.
(3) There are no rectangles.
(4) Given a line
l
P
l
, there are infinitely many lines through
P
l
(5) Similar triangles are congruent.
Proofs: The proof of (1) is lengthy and is omitted for now.
Clearly (1) implies (2), since a quadrilateral can be split into two triangles by drawing a diagonal. And (2) implies (3), since a rectangle is a quadrilateral with all right angles.
To see (4), drop a perpendicular from
Q
l
; say it meets
l
Q
. (It is convenient to use lower-case letters for lines, and upper-case letters for points.) Then draw a perpendicular line
m
PQ
with right angle at
P
. It's impossible for
l
m
to meet, because if they met at a point
X
PQX
would be a triangle with two right angles, which would contradict (1). So
m
l
Source: http://www.math.cornell.edu/~mec/Winter2009/Mihai/section5.html
Pick a different point
R
L
, draw a line
n
l
R
, and drop a perpendicular
PS
n
PS
l
by the same two-right-angles argument as above, but
PS
is not the same line as
m
; if it were, then
PQRS
would be a rectangle. For any
R
l
, we thus get a new line
PS
l
, and a similar argument shows that different choices of
R
lead to different lines.
To see (5), suppose the triangles are as pictured. Find a triangle
AB''C''
congruent to the smaller one lying inside the bigger one. Source: http://www.math.cornell.edu/~mec/Winter2009/Mihai/section5.html But then it is straightforward to see that the two angles on the left side of the quadrilateral
BB''C''C
180^\circ
, and similarly for the two angles on the right side, so the angles sum to
360^\circ
, which contradicts (2).
There are two common models used to picture lines and angles in plane (two-dimensional) hyperbolic geometry. They are both due to Poincare.
Poincare half-plane model: The points are the complex numbers in the set
{\mathbb H} = \{ z \in {\mathbb C} : \text{Im}(z) > 0 \}
. This is called the upper half-plane; in Cartesian coordinates it consists of points
(x,y)
y
Lines in this model are of two types: straight vertical lines, and half-circles whose centers lie on the
x
-axis. Angles are measured as one would expect (the angle between the two curves at the point of intersection), but distances are trickier. Generally speaking, one thinks of distances between points near the
x
-axis as "blowing up"; one way to represent this is that the infinitesimal unit of distance
ds
satisfies the formula
(ds)^2 = \frac{(dx)^2+(dy)^2}{y^2},
rather than the usual
(ds)^2 = (dx)^2+(dy)^2
in standard analysis. There is a formula for the distance between two points
z
w
that uses the inverse hyperbolic trigonometric functions, similar to the one in the Poincare disk model (see below), but it is unwieldy to work with. (One way to think of this is that it is the price one pays for keeping angles at their normal values; there is another model due to Beltrami and Klein with a nicer distance function, and lines which are straight, but angle measures in this model are distorted.)
CD and EF are parallel, EF and GH are parallel, but CD and GH are not.
Poincare disk model: Using a conformal mapping that takes the
x
-axis to the unit circle gives a model of hyperbolic geometry contained inside the unit disk. In this model, lines are either diameters of the disk or the intersection of a circle
C
with the disk, where
C
is perpendicular to the unit circle at its two points of intersection. Angles continue to be measured as expected.
The distance formula for two complex numbers
z,w
inside the disk becomes
d(z,w) = \text{arccosh}\left( 1+2\frac{|z-w|^2}{\big(1-|z|^2\big)\big(1-|w|^2\big)} \right),
\text{arccosh}(x) = \ln\big(x+\sqrt{x^2-1}\big)
is the inverse function of the hyperbolic cosine.
|z|
1
, the distance between
z
and another point goes to
\infty
This is a tiling of the hyperbolic plane by congruent triangles.
Modular forms are fundamental objects in modern number theory; they were famously used in Wiles' proof of Fermat's last theorem. They are functions defined on the upper half-plane which are invariant under isometry, which is a (hyperbolic-) distance-preserving map from the upper half-plane to itself. So properties of hyperbolic geometry become important when studying modular forms.
There is an explicit connection between special relativity and hyperbolic geometry via Minkowski space, which is a common setting for both of them.
Hyperbolic geometry graphs have been suggested as a promising model for social networks where the hyperbolicity appears through a competition between similarity and popularity of an individual.
The artist M.C. Escher created many beautiful artworks based on tessellations of the Poincare unit disk. Both "Circle Limit III" and "Circle Limit IV" are famous examples.
HyperRogue is a computer game that lets you experience hyperbolic geometry. It plays similar to a single-player boardgame, but where the chessboard is randomly generated by the computer. In case of HyperRogue, the chessboard is an infinite hyperbolic plane. Many of strategies, locations, and navigation puzzles in HyperRogue are based on the properties of the hyperbolic plane.
Cite as: Hyperbolic Geometry. Brilliant.org. Retrieved from https://brilliant.org/wiki/hyperbolic-geometry/
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Asymptotic analysis - Wikipedia
Description of limiting behavior of a function
This article is about the behavior of functions as inputs approach infinity or some other limit value. For asymptotes in geometry, see Asymptote.
As an illustration, suppose that we are interested in the properties of a function f (n) as n becomes very large. If f(n) = n2 + 3n, then as n becomes very large, the term 3n becomes insignificant compared to n2. The function f(n) is said to be "asymptotically equivalent to n2, as n → ∞". This is often written symbolically as f (n) ~ n2, which is read as "f(n) is asymptotic to n2".
An example of an important asymptotic result is the prime number theorem. Let π(x) denote the prime-counting function (which is not directly related to the constant pi), i.e. π(x) is the number of prime numbers that are less than or equal to x. Then the theorem states that
{\displaystyle \pi (x)\sim {\frac {x}{\ln x}}.}
Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation.
3 Examples of asymptotic formulas
4.2 Asymptotic to two different polynomials
5 Asymptotic expansion
5.1 Examples of asymptotic expansions
6 Asymptotic distribution
Formally, given functions f (x) and g(x), we define a binary relation
{\displaystyle f(x)\sim g(x)\quad ({\text{as }}x\to \infty )}
if and only if (de Bruijn 1981, §1.4)
{\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=1.}
The symbol ~ is the tilde. The relation is an equivalence relation on the set of functions of x; the functions f and g are said to be asymptotically equivalent. The domain of f and g can be any set for which the limit is defined: e.g. real numbers, complex numbers, positive integers.
The same notation is also used for other ways of passing to a limit: e.g. x → 0, x ↓ 0, |x| → 0. The way of passing to the limit is often not stated explicitly, if it is clear from the context.
Although the above definition is common in the literature, it is problematic if g(x) is zero infinitely often as x goes to the limiting value. For that reason, some authors use an alternative definition. The alternative definition, in little-o notation, is that f ~ g if and only if
{\displaystyle f(x)=g(x)(1+o(1)).}
This definition is equivalent to the prior definition if g(x) is not zero in some neighbourhood of the limiting value.[1][2]
{\displaystyle f\sim g}
{\displaystyle a\sim b}
, then under some mild conditions,[further explanation needed] the following hold:
{\displaystyle f^{r}\sim g^{r}}
, for every real r
{\displaystyle \log(f)\sim \log(g)}
{\displaystyle \lim g\neq 1}
{\displaystyle f\times a\sim g\times b}
{\displaystyle f/a\sim g/b}
Such properties allow asymptotically-equivalent functions to be freely exchanged in many algebraic expressions.
Examples of asymptotic formulas[edit]
{\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}
—this is Stirling's approximation
For a positive integer n, the partition function, p(n), gives the number of ways of writing the integer n as a sum of positive integers, where the order of addends is not considered.
{\displaystyle p(n)\sim {\frac {1}{4n{\sqrt {3}}}}e^{\pi {\sqrt {\frac {2n}{3}}}}}
The Airy function, Ai(x), is a solution of the differential equation y″ − xy = 0; it has many applications in physics.
{\displaystyle \operatorname {Ai} (x)\sim {\frac {e^{-{\frac {2}{3}}x^{\frac {3}{2}}}}{2{\sqrt {\pi }}x^{1/4}}}}
{\displaystyle {\begin{aligned}H_{\alpha }^{(1)}(z)&\sim {\sqrt {\frac {2}{\pi z}}}e^{i\left(z-{\frac {2\pi \alpha -\pi }{4}}\right)}\\H_{\alpha }^{(2)}(z)&\sim {\sqrt {\frac {2}{\pi z}}}e^{-i\left(z-{\frac {2\pi \alpha -\pi }{4}}\right)}\end{aligned}}}
{\displaystyle h(x)=f(x)(1-F(x))+g(x)F(x)}
{\displaystyle f(x)}
{\displaystyle g(x)}
are real-valued analytic functions, and
{\displaystyle F(x)}
is a Cumulative distribution function.
{\displaystyle h(x)}
{\displaystyle f(x)}
{\displaystyle x\to (-\infty )}
and asymptotic to
{\displaystyle g(x)}
{\displaystyle x\to (+\infty )}
Asymptotic to two different polynomials[edit]
Suppose we want a real-valued function that is asymptotic to
{\displaystyle (a_{0}+a_{1}x)}
{\displaystyle x\to (-\infty )}
and is asymptotic to
{\displaystyle (b_{0}+b_{1}x)}
{\displaystyle x\to (+\infty )}
{\displaystyle h(x)=(a_{0}+a_{1}x)(1-F(x))+(b_{0}+b_{1}x)F(x)}
Main article: Asymptotic expansion
An asymptotic expansion of a function f(x) is in practice an expression of that function in terms of a series, the partial sums of which do not necessarily converge, but such that taking any initial partial sum provides an asymptotic formula for f. The idea is that successive terms provide an increasingly accurate description of the order of growth of f.
In symbols, it means we have
{\displaystyle f\sim g_{1},}
{\displaystyle f-g_{1}\sim g_{2}}
{\displaystyle f-g_{1}-\cdots -g_{k-1}\sim g_{k}}
for each fixed k. In view of the definition of the
{\displaystyle \sim }
symbol, the last equation means
{\displaystyle f-(g_{1}+\cdots +g_{k})=o(g_{k})}
in the little o notation, i.e.,
{\displaystyle f-(g_{1}+\cdots +g_{k})}
{\displaystyle g_{k}.}
{\displaystyle f-g_{1}-\cdots -g_{k-1}\sim g_{k}}
takes its full meaning if
{\displaystyle g_{k+1}=o(g_{k})}
for all k, which means the
{\displaystyle g_{k}}
form an asymptotic scale. In that case, some authors may abusively write
{\displaystyle f\sim g_{1}+\cdots +g_{k}}
to denote the statement
{\displaystyle f-(g_{1}+\cdots +g_{k})=o(g_{k}).}
One should however be careful that this is not a standard use of the
{\displaystyle \sim }
symbol, and that it does not correspond to the definition given in § Definition.
In the present situation, this relation
{\displaystyle g_{k}=o(g_{k-1})}
actually follows from combining steps k and k−1; by subtracting
{\displaystyle f-g_{1}-\cdots -g_{k-2}=g_{k-1}+o(g_{k-1})}
{\displaystyle f-g_{1}-\cdots -g_{k-2}-g_{k-1}=g_{k}+o(g_{k}),}
{\displaystyle g_{k}+o(g_{k})=o(g_{k-1}),}
{\displaystyle g_{k}=o(g_{k-1}).}
In case the asymptotic expansion does not converge, for any particular value of the argument there will be a particular partial sum which provides the best approximation and adding additional terms will decrease the accuracy. This optimal partial sum will usually have more terms as the argument approaches the limit value.
Examples of asymptotic expansions[edit]
{\displaystyle {\frac {e^{x}}{x^{x}{\sqrt {2\pi x}}}}\Gamma (x+1)\sim 1+{\frac {1}{12x}}+{\frac {1}{288x^{2}}}-{\frac {139}{51840x^{3}}}-\cdots \ (x\to \infty )}
{\displaystyle xe^{x}E_{1}(x)\sim \sum _{n=0}^{\infty }{\frac {(-1)^{n}n!}{x^{n}}}\ (x\to \infty )}
{\displaystyle {\sqrt {\pi }}xe^{x^{2}}\operatorname {erfc} (x)\sim 1+\sum _{n=1}^{\infty }(-1)^{n}{\frac {(2n-1)!!}{n!(2x^{2})^{n}}}\ (x\to \infty )}
where m!! is the double factorial.
Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. For example, we might start with the ordinary series
{\displaystyle {\frac {1}{1-w}}=\sum _{n=0}^{\infty }w^{n}}
The expression on the left is valid on the entire complex plane
{\displaystyle w\neq 1}
, while the right hand side converges only for
{\displaystyle |w|<1}
. Multiplying by
{\displaystyle e^{-w/t}}
and integrating both sides yields
{\displaystyle \int _{0}^{\infty }{\frac {e^{-{\frac {w}{t}}}}{1-w}}\,dw=\sum _{n=0}^{\infty }t^{n+1}\int _{0}^{\infty }e^{-u}u^{n}\,du}
The integral on the left hand side can be expressed in terms of the exponential integral. The integral on the right hand side, after the substitution
{\displaystyle u=w/t}
, may be recognized as the gamma function. Evaluating both, one obtains the asymptotic expansion
{\displaystyle e^{-{\frac {1}{t}}}\operatorname {Ei} \left({\frac {1}{t}}\right)=\sum _{n=0}^{\infty }n!\;t^{n+1}}
Here, the right hand side is clearly not convergent for any non-zero value of t. However, by keeping t small, and truncating the series on the right to a finite number of terms, one may obtain a fairly good approximation to the value of
{\displaystyle \operatorname {Ei} (1/t)}
{\displaystyle x=-1/t}
{\displaystyle \operatorname {Ei} (x)=-E_{1}(-x)}
results in the asymptotic expansion given earlier in this article.
Asymptotic distribution[edit]
Main article: Asymptotic distribution
In mathematical statistics, an asymptotic distribution is a hypothetical distribution that is in a sense the "limiting" distribution of a sequence of distributions. A distribution is an ordered set of random variables Zi for i = 1, …, n, for some positive integer n. An asymptotic distribution allows i to range without bound, that is, n is infinite.
A special case of an asymptotic distribution is when the late entries go to zero—that is, the Zi go to 0 as i goes to infinity. Some instances of "asymptotic distribution" refer only to this special case.
This is based on the notion of an asymptotic function which cleanly approaches a constant value (the asymptote) as the independent variable goes to infinity; "clean" in this sense meaning that for any desired closeness epsilon there is some value of the independent variable after which the function never differs from the constant by more than epsilon.
An asymptote is a straight line that a curve approaches but never meets or crosses. Informally, one may speak of the curve meeting the asymptote "at infinity" although this is not a precise definition. In the equation
{\displaystyle y={\frac {1}{x}},}
y becomes arbitrarily small in magnitude as x increases.
Asymptotic analysis is used in several mathematical sciences. In statistics, asymptotic theory provides limiting approximations of the probability distribution of sample statistics, such as the likelihood ratio statistic and the expected value of the deviance. Asymptotic theory does not provide a method of evaluating the finite-sample distributions of sample statistics, however. Non-asymptotic bounds are provided by methods of approximation theory.
Examples of applications are the following.
In applied mathematics, asymptotic analysis is used to build numerical methods to approximate equation solutions.
In mathematical statistics and probability theory, asymptotics are used in analysis of long-run or large-sample behaviour of random variables and estimators.
In computer science in the analysis of algorithms, considering the performance of algorithms.
The behavior of physical systems, an example being statistical mechanics.
In accident analysis when identifying the causation of crash through count modeling with large number of crash counts in a given time and space.
Asymptotic analysis is a key tool for exploring the ordinary and partial differential equations which arise in the mathematical modelling of real-world phenomena.[3] An illustrative example is the derivation of the boundary layer equations from the full Navier-Stokes equations governing fluid flow. In many cases, the asymptotic expansion is in power of a small parameter, ε: in the boundary layer case, this is the nondimensional ratio of the boundary layer thickness to a typical length scale of the problem. Indeed, applications of asymptotic analysis in mathematical modelling often[3] center around a nondimensional parameter which has been shown, or assumed, to be small through a consideration of the scales of the problem at hand.
Asymptotic expansions typically arise in the approximation of certain integrals (Laplace's method, saddle-point method, method of steepest descent) or in the approximation of probability distributions (Edgeworth series). The Feynman graphs in quantum field theory are another example of asymptotic expansions which often do not converge.
Asymptotic density (in number theory)
Asymptotology
Leading-order term
Method of dominant balance (for ODEs)
Watson's lemma
^ "Asymptotic equality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
^ Estrada & Kanwal (2002, §1.2)
^ a b Howison, S. (2005), Practical Applied Mathematics, Cambridge University Press
Balser, W. (1994), From Divergent Power Series To Analytic Functions, Springer-Verlag, ISBN 9783540485940
de Bruijn, N. G. (1981), Asymptotic Methods in Analysis, Dover Publications, ISBN 9780486642215
Estrada, R.; Kanwal, R. P. (2002), A Distributional Approach to Asymptotics, Birkhäuser, ISBN 9780817681302
Miller, P. D. (2006), Applied Asymptotic Analysis, American Mathematical Society, ISBN 9780821840788
Murray, J. D. (1984), Asymptotic Analysis, Springer, ISBN 9781461211228
Paris, R. B.; Kaminsky, D. (2001), Asymptotics and Mellin-Barnes Integrals, Cambridge University Press
Asymptotic Analysis —home page of the journal, which is published by IOS Press
A paper on time series analysis using asymptotic distribution
Retrieved from "https://en.wikipedia.org/w/index.php?title=Asymptotic_analysis&oldid=1073726568"
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