A numerical wavefront solution for quantum transmission lines with charge discreteness is
proposed for the first time. The nonlinearity of the system becomes deeply related to charge discreteness. The
wavefront velocity is found to depend on the normalized (pseudo) flux variable. Finally we find the dispersion
relation for the normalized flux
0 / .
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MATLAB sessions: Laboratory 6
MAT 275 Laboratory 6
Forced Equations and Resonance
In this laboratory we take a deeper look at second-order nonhomogeneous equations. We will concentrate
on equations with a periodic harmonic forcing term. This will lead to a study of the phenomenon known
as resonance. The equation we consider has the form
d2y
dt2
+ c
dy
dt
+ ω20y = cosωt. (L6.1)
This equation models the movement of a mass-spring system similar to the one described in Laboratory
5. The forcing term on the right-hand side of (L6.1) models a vibration, with amplitude 1 and frequency
ω (in radians per second = 12π rotation per second =
60
2π rotations per minute, or RPM) of the plate
holding the mass-spring system. All physical constants are assumed to be positive.
Let ω1 =
√
ω20 − c2/4. When c < 2ω0 the general solution of (L6.1) is
y(t) = e−
1
2 ct(c1 cos(ω1t) + c2 sin(ω1t)) + C cos (ωt− α) (L6.2)
with
C =
1√
(ω20 − ω2)
2
+ c2ω2
, (L6.3)
α =
⎧
⎨
⎩
arctan
(
cω
ω20−ω2
)
if ω0 > ω
π + arctan
(
cω
ω20−ω2
)
if ω0 < ω
(L6.4)
and c1 and c2 determined by the initial conditions. The first term in (L6.2) represents the complementary
solution, that is, the general solution to the homogeneous equation (independent of ω), while the second
term represents a particular solution of the full ODE.
Note that when c > 0 the first term vanishes for large t due to the decreasing exponential factor.
The solution then settles into a (forced) oscillation with amplitude C given by (L6.3). The objectives of
this laboratory are then to understand
1. the effect of the forcing term on the behavior of the solution for different values of ω, in particular
on the amplitude of the solution.
2. the phenomena of resonance and beats in the absence of friction.
The Amplitude of Forced Oscillations
We assume here that ω0 = 2 and c = 1 are fixed. Initial conditions are set to 0. For each value of ω, the
amplitude C can be obtained numerically by taking half the difference between the highs and the lows
of the solution computed with a MATLAB ODE solver after a sufficiently large time, as follows: (note
that in the M-file below we set ω = 1.4).
1 function LAB06ex1
2 omega0 = 2; c = 1; omega = 1.4;
3 param = [omega0,c,omega];
4 t0 = 0; y0 = 0; v0 = 0; Y0 = [y0;v0]; tf = 50;
5 options = odeset(’AbsTol’,1e-10,’RelTol’,1e-10);
6 [t,Y] = ode45(@f,[t0,tf],Y0,options,param);
7 y = Y(:,1); v = Y(:,2);
8 figure(1)
9 plot(t,y,’b-’); ylabel(’y’); grid on;
c⃝2011 Stefania Tracogna, SoMSS, ASU 1
MATLAB sessions: Laboratory 6
10 t1 = 25; i = find(t>t1);
11 C = (max(Y(i,1))-min(Y(i,1)))/2;
12 disp([’computed amplitude of forced oscillation = ’ num2str(C)]);
13 Ctheory = 1/sqrt((omega0^2-omega^2)^2+(c*omega)^2);
14 disp([’theoretical amplitude = ’ num2str(Ctheory)]);
15 %----------------------------------------------------------------
16 function dYdt = f(t,Y,param)
17 y = Y(1); v = Y(2);
18 omega0 = param(1); c = param(2); omega = param(3);
19 dYdt = [ v ; cos(omega ...
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Artificial Neural Network Basics and their Working in random sample of inputs
Here explain Basic Implementation of Artificial Neural network
in python aJupyter
Hec registration form VISUAL C# PROGRAMMINGAYESHA JAVED
VISUAL PROGRAMING REGISTRATION FOAM CODE +SCREENSHOT OF THE OUTPUT
Visual C# Programming
HEC REGISTRATION FORM
HELPFUL IN WRITING CODE UNDERSTAND THE VISUAL C#
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
APPLICATION OF HIGHER ORDER DIFFERENTIAL EQUATIONS
1. MODELING OF SECOND ORDER
DIFFERENTIAL EQUATION
And Applications of Second
Order Differential Equations:-
2. GROUP MEMBERS
AYESHA JAVED(30)
SAFEENA AFAQ(26)
RABIA AZIZ(40)
SHAMAIN FATIMA(50)
UMAIRA ZIA(35)
3.
4. MODELING
A mathematical model is a description of a system using
mathematical concepts and language. The process of developing a
mathematical model is termed as Mathematical modeling.
Mathematical models are used in the
Natural sciences
(such as physics, biology, earth science, chemistry)
Engineering disciplines
(such as computer science, electrical engineering)
Social sciences
(such as economics, psychology, sociology, political science).
5. EXPLANATION
A model may help to explain a system and to study the
effects of different components, and to make predictions
about behavior.
6.
7. SECOND ORDER DIFFERENTIAL EQUATION
A second order differential equation is an equation
involving the unknown function y, its derivatives y'
and y'', and the variable x. We will consider explicit
differential equations of the form:
Explicit solution is a solution where the dependent
variable can be separated.
Where a, b, and c are constants.
8. HOMOGENOUS EQUATION
The differential equation:
Is a second order, constant coefficient, linear, homogenous
differential equation. Its solution is found from the solutions
to the auxiliary equation:
aD2+bD+c=0
These are:
9. AUXILIARY EQUATION
The characteristic equation (or auxiliary equation) is
an algebraic equation of degree n upon which depends
the solution of a given nth-order differential equation
or difference equation. The characteristic equation can
only be formed when the differential or difference
equation is linear and homogeneous, and has
constant coefficients. There are three types of roots:
Real and distinct roots
Real and repeater roots
Complex roots
10. REAL AND DISTINCT:
If the Auxiliary equation:
am2+bm+c=0
With solution:
Where:
m1and m2 are real and m1≠ m2
Then the solution is:
11. REAL AND REPEATED:
If the Auxiliary equation:
am2+bm+c=0
With solution:
Where:
m1and m2 are real and m1=m2
Then the solution is:
12. COMPLEX ROOTS
If the Auxiliary equation:
am2+bm+c=0
With solution:
Where:
m1and m2 are complex
Then the solution to differential equation whose roots are
complex can be written as:
13.
14. NON-HOMOGENOUS EQUATION
The second order, constant coefficient, linear, non-
homogenous differential equation is an equation of the
type:
The solution is in two parts yc+ yp:
a) Part1, yc is the solution to the homogenous equation
and is called complementary function which is the
solution to the homogenous equation
b) Part 2, yp is called the particular integral.
15. f(t) Choice for yp(t)
a xn cnxn+cn-1xn-1+…+c1x1+c0
a xneat eat (cnxn+cn-1xn-1+…+c1x1+c0 )
K sin(at) c1 sin(at)+c2 cos(at)
K cos(at) c1 sin(at)+c2 cos(at)
K eat sin(bt) eat (c1 sin(bt)+c2 cos(bt))
K eat cos(bt) eat (c1 sin(bt)+c2 cos(bt))
TABLE
17. Particular integer: (part 2)
Assume a form for yp as yp=Ax2 +Bx +C,
dyp/dx =2Ax +B
d2yp/dx2 =2A
then substitution in:
We get: 2A -10Ax -5B +6Ax2 +6Bx +6C = x2
6Ax2 +(6B-10A)x +(6C+2A-5B)=x2 +0x +0
comparing coefficients of x2, x and constant:
A=1/6; B=5/18; C=19/18
So that:
yp =(1/6)x2 +(5/18)x +(19/18)
18. The Complete solution of:
Consist of:
Complementary function+ particular function
That is:
y= yc+yp = c1e2x +c2e3x +(1/6)x2 +(5/18)x +(19/18)
19.
20. APPLICATIONS OF SECOND ORDER DIFFERENTIAL
EQUATION:
Second-order linear differential equations have a variety of
applications in science and engineering. In this section we
explore two of them:
1) The vibration of springs
2) Electric current circuits.
24. The force exerted by a spring is given by Hooke's
Law; this states that if a spring is stretched or
compressed a distance x from its natural length,
then it exerts a force given by the equation
HOOKE’S LAW
25. VIBRATING SPRINGS
According to Hooke’s Law, which says
that if the spring is stretched (or
compressed) units from its natural
length, then it exerts a force that is
proportional to :
restoring force= -kx
26. EXPLANATION:
Where k is a positive constant (called the spring constant) and By
Newton’s Second Law, we have:
F = ma = -kx
Acceleration is defined as the rate of change of velocity and
velocity of an object is the rate of change of its position with
respect to time t.
a= dv/dt ;
v= dx/dt ;
a= d2x/dt2
By substituting the value, we have:
m(d2x/dt2)= -kx
27. CONTINUE…
We get:
This is second-order linear differential equation. Its characteristic
equation is mD2 +k =0
D2 =-k/m;
D=√-k/m;
Its roots are: We can combine the constants k and m by making the
substitution: √ k/m= ω
D = ±ωi
28. This type of motion is called simple harmonic motion.
As r=d2x/dt2 The solution of Eq. is a function of time
D = ±ωi
a ±bi ; a=0 and b= ω The solution of Eq. (1) is a function of time
x(t) =eat (c1 cos(b)+ c2 sin(b))
Thus, the general solution is:
x(t) =e0t (c1 cos(ωt)+ c2 sin(ωt))
x(t) =c1 cos(ωt)+ c2 sin(ωt))
AS ω = √ k/m
29. EXAMPLE
A spring with a mass of 2 kg has natural length 0.5 m. A force of
25.6 N is required to maintain it stretched to a length of 0.7 m. If
the spring is stretched to a length of 0.7 m and then released with
initial velocity 0, find the position of the mass at any time t.
Solution:
From Hooke’s Law, the force required to stretch the spring is
k (0.2)=25.6
k =25.5/0.2 =128
So, Using this value of the spring constant , together with m=2,
we have
30. As in the earlier general discussion, the solution of this equation
is
2D2 +128 =0
D2 =-128/2 = -64
D = ± 8i
Thus, the general solution is:
x(t) =c1 cos(8t)+ c2 sin(8t)
We are given the initial condition that x(0)= 0.2. But from above
equation, we have:
x(0) =c1 cos(0)+ c2 sin(0) = c1
Comparing both we get:
c1 = 0.2
31. Differentiate equation, we get:
x’(t) = -8c1 sin(8t)+ 8c2 cos(8t)
Since the initial velocity is given as x’(0)=0, and
x’(0) = -8c1 sin(0)+ 8c2 cos(0)
x’(0) = 8c2
Compare both equations, we get:
c2 = 0
so the solution is:
x(t) =(1/5) cos (8t)
Or
x(t) =(0.2) cos (8t)
32.
33. ELECTRIC CURRENT CIRCUIT
The simple electrical circuit consists of a resistor R in ohms; a
capacitor C in farads; an inductor L in henries ; and an
electromotive force (emf) E(t) in volts , usually a battery or a
generator, all connected in series. The current I flowing
through the circuit is measured in amperes and the charge Q on
the capacitor is measured in coulombs.
34. EXPLANATION
It is known that the voltage drops across a resistor, a capacitor,
and an inductor are respectively RI, (1/C)Q, and L(dI/dt)
where Q is the charge on the capacitor. The voltage drop
across an emf is E(t). Thus, from Kirchhoff's loop law, we
have:
1
36. CONTINUE…
The relationship between I and Q is
I = dQ/dt, dI/dt = d2Q/ dt2
Substituting these values into the above equation, we
obtain
which is a second-order linear differential equation with
constant coefficients. If the charge Q and the current I
are known at time 0, then we have the initial condition
37. Thus
Q(0) = Q0 , (dQ/dt)|t=0 =I(0) = I0
To obtain a differential equation for the current, we differentiate
equation (1) with respect to t and then substitute it directly into
the resulting equation to obtain
The first initial condition is
I(0)= I0
38. The second initial condition is obtained from equation (1) by
solving for dI/dt and then setting t=0. Thus
Eq 1
L dI/dt=E(t)-RI-Q/C
(dI/dt)= (1/L)E(t) –(R/L)I –(1/LC )Q
NOW t=0 and I=I0 , Q= Q0
(dI/dt)|t=0 = (1/L)E(0) –(R/L)I0 –(1/LC )Q0
41. EXAMPLE
Find the charge and current at time in the circuit if R=40Ὠ,
L= 1H, C= 16X 10-4F, E(t)= 100cos 10t, and the initial
charge and current are both 0.
Solution:
With the given values of L, R, C, and E(t), it becomes:
(Part 1) The auxiliary equation is r2+40r+625= 0 with
roots
42. SOLUTION:
so the solution of the complementary equation is
Qc (t) =e-20t(c1cos 15t+c2sin 15t)
(Part 2) For the method of undetermined coefficients we try the
particular solution
Qp (t)= Acos 10t+Bsin 10t
Qp'(t) = -10Asin10t +10Bcos 10t
Qp" (t) = -100Acos 10t -100Bsin 10t
Substituting it in first equation, we have
(-100Acos 10t -100Bsin 10t ) +40(-10Asin10t +10Bcos 10t)
+625(Acos 10t+Bsin 10t) = 100cos 10t
43. (525A + 400B) cos 10t +(-400A +525B)sin 10t = 100cos 10t
The solution of this system is
A = 84/697; B =64/697
so a particular solution is
Qp (t)= (84/697)cos 10t + (64/697)sin 10t)
and the general solution is
Q (t) =Qc (t) +Qp (t) =e-20t(c1cos 15t+c2sin 15t) + (84/697)cos 10t
+(64/697)sin 10t
Imposing the initial condition Q (0)=0 , we get
Q (0)= c1 +(84/697) =0
c1 = -84/697
44. To impose the other initial condition we first differentiate to find the
current:
I = Qp'(t)=e-20t(-15c1sin15t+15c2cos15t -20 e-20t(c1cos 15t+c2sin 15t) -
10(84/697)sin 10t +10(64/697)cos 10t
I = e-20t[-15c1sin15t+15c2cos15t -20c1cos 15t -20c2sin 15t]
+(40/697)[-21 sin 10t +16 cos 10t]
I = e-20t[(-20c1 +15c2)cos 15t +(-15c1 -20c2)sin 15t ] +(40/697)[-21 sin
10t +16 cos 10t]
I (0) = -20c1 +15c2 +(640/697) = 0
c2 = -(464/2091)
45. Thus, the formula for the charge Q is:
and the expression for the current I is