A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
Higher braiding gates, a new kind of quantum gate, are introduced. These are matrix solutions of the polyadic braid equations (which differ from the generalized Yang-Baxter equations). Such gates support a special kind of multi-qubit entanglement which can speed up key distribution and accelerate the execution of algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates which can be related to qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, the star and circle types, and find that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the classes introduced here is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by higher braid operators are given. Finally, we show that for each multi-qubit state there exist higher braiding gates which are not entangling, and the concrete conditions to be non-entangling are given for the binary and ternary gates discussed.
The Gaussian or normal distribution is one of the most widely used in statistics. Estimating its parameters using
Bayesian inference and conjugate priors is also widely used. The use of conjugate priors allows all the results to be
derived in closed form. Unfortunately, different books use different conventions on how to parameterize the various
distributions (e.g., put the prior on the precision or the variance, use an inverse gamma or inverse chi-squared, etc),
which can be very confusing for the student. In this report, we summarize all of the most commonly used forms. We
provide detailed derivations for some of these results; the rest can be obtained by simple reparameterization. See the
appendix for the definition the distributions that are used.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few more good examples.
A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
Higher braiding gates, a new kind of quantum gate, are introduced. These are matrix solutions of the polyadic braid equations (which differ from the generalized Yang-Baxter equations). Such gates support a special kind of multi-qubit entanglement which can speed up key distribution and accelerate the execution of algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates which can be related to qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, the star and circle types, and find that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the classes introduced here is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by higher braid operators are given. Finally, we show that for each multi-qubit state there exist higher braiding gates which are not entangling, and the concrete conditions to be non-entangling are given for the binary and ternary gates discussed.
The Gaussian or normal distribution is one of the most widely used in statistics. Estimating its parameters using
Bayesian inference and conjugate priors is also widely used. The use of conjugate priors allows all the results to be
derived in closed form. Unfortunately, different books use different conventions on how to parameterize the various
distributions (e.g., put the prior on the precision or the variance, use an inverse gamma or inverse chi-squared, etc),
which can be very confusing for the student. In this report, we summarize all of the most commonly used forms. We
provide detailed derivations for some of these results; the rest can be obtained by simple reparameterization. See the
appendix for the definition the distributions that are used.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few more good examples.
Lesson 16: Inverse Trigonometric Functions (Section 041 slides)Mel Anthony Pepito
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Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few more good examples.
Mat 121-Limits education tutorial 22 I.pdfyavig57063
limitsExample: A function C=f(d) gives the number of classes
C, a student takes in a day, d of the week. What does
f(Monday)=4 mean?
Solution. From f(Monday)=4, we see that the input day
is Monday while the output value, number of courses is
4. Thus, the student takes 4 classes on Mondays.Function: is a rule which assigns an element in
the domain to an element in the range in such a
way that each element in the domain
corresponds to exactly one element in the range.
The notation f(x) read “f of x” or “f at x” means
function of x while the notion y=f(x) means y is a
function of x. The letter x represents the input
value, or independent variable
In this paper, optimal control problem for processes represented by stochastic sequential machine is
analyzed. Principle of optimality is proven for the considered problem. Then by using method of dynamical
programming, solution of optimal control problem is found.
Opening of our Deep Learning Lunch & Learn series. First session: introduction to Neural Networks, Gradient descent and backpropagation, by Pablo J. Villacorta, with a prologue by Fernando Velasco
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
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• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
RAT: Retrieval Augmented Thoughts Elicit Context-Aware Reasoning in Long-Hori...
Numerical analysis m2 l4slides
1. D Nagesh Kumar, IISc Optimization Methods: M2L41
Optimization using Calculus
Optimization of Functions of
Multiple Variables subject to
Equality Constraints
2. D Nagesh Kumar, IISc Optimization Methods: M2L42
Objectives
Optimization of functions of multiple variables
subjected to equality constraints using
the method of constrained variation, and
the method of Lagrange multipliers.
3. D Nagesh Kumar, IISc Optimization Methods: M2L43
Constrained optimization
A function of multiple variables, f(x), is to be optimized subject to one or
more equality constraints of many variables. These equality constraints,
gj(x), may or may not be linear. The problem statement is as follows:
Maximize (or minimize) f(X), subject to gj(X) = 0, j = 1, 2, … , m
where
1
2
n
x
x
x
⎧ ⎫
⎪ ⎪
⎪ ⎪
= ⎨ ⎬
⎪ ⎪
⎪ ⎪⎩ ⎭
X
M
4. D Nagesh Kumar, IISc Optimization Methods: M2L44
Constrained optimization (contd.)
With the condition that ; or else if m > n then the problem
becomes an over defined one and there will be no solution. Of the
many available methods, the method of constrained variation and the
method of using Lagrange multipliers are discussed.
m n≤
5. D Nagesh Kumar, IISc Optimization Methods: M2L45
Solution by method of Constrained
Variation
For the optimization problem defined above, let us consider a specific
case with n = 2 and m = 1 before we proceed to find the necessary and
sufficient conditions for a general problem using Lagrange
multipliers. The problem statement is as follows:
Minimize f(x1,x2), subject to g(x1,x2) = 0
For f(x1,x2) to have a minimum at a point X* = [x1
*,x2
*], a necessary
condition is that the total derivative of f(x1,x2) must be zero at [x1
*,x2
*].
(1)
1 2
1 2
0
f f
df dx dx
x x
∂ ∂
= + =
∂ ∂
6. D Nagesh Kumar, IISc Optimization Methods: M2L46
Method of Constrained Variation (contd.)
Since g(x1
*,x2
*) = 0 at the minimum point, variations dx1 and dx2 about
the point [x1
*, x2
*] must be admissible variations, i.e. the point lies on
the constraint:
g(x1
* + dx1 , x2
* + dx2) = 0 (2)
assuming dx1 and dx2 are small the Taylor series expansion of this
gives us
(3)
*
1 1 2 2
* * * * * *
1 2 1 2 1 1 2 2
1 2
( * , )
( , ) (x ,x ) (x ,x ) 0
g x dx x dx
g g
g x x dx dx
x x
+ +
∂ ∂
= + + =
∂ ∂
7. D Nagesh Kumar, IISc Optimization Methods: M2L47
Method of Constrained Variation (contd.)
or at [x1
*,x2
*] (4)
which is the condition that must be satisfied for all admissible
variations.
Assuming , (4) can be rewritten as
(5)
1 2
1 2
0
g g
dg dx dx
x x
∂ ∂
= + =
∂ ∂
2/ 0g x∂ ∂ ≠
* *1
2 1 2 1
2
/
( , )
/
g x
dx x x dx
g x
∂ ∂
= −
∂ ∂
8. D Nagesh Kumar, IISc Optimization Methods: M2L48
Method of Constrained Variation (contd.)
(5) indicates that once variation along x1 (dx1) is chosen arbitrarily, the variation
along x2 (dx2) is decided automatically to satisfy the condition for the admissible
variation. Substituting equation (5) in (1) we have:
(6)
The equation on the left hand side is called the constrained variation of f. Equation
(5) has to be satisfied for all dx1, hence we have
(7)
This gives us the necessary condition to have [x1
*, x2
*] as an extreme point
(maximum or minimum)
* *
1 2
1
1
1 2 2 (x , x )
/
0
/
f g x f
df dx
x g x x
⎛ ⎞∂ ∂ ∂ ∂
= − =⎜ ⎟
∂ ∂ ∂ ∂⎝ ⎠
* *
1 2
1 2 2 1 (x , x )
0
f g f g
x x x x
⎛ ⎞∂ ∂ ∂ ∂
− =⎜ ⎟
∂ ∂ ∂ ∂⎝ ⎠
9. D Nagesh Kumar, IISc Optimization Methods: M2L49
Solution by method of Lagrange
multipliers
Continuing with the same specific case of the optimization problem
with n = 2 and m = 1 we define a quantity λ, called the Lagrange
multiplier as
(8)
Using this in (5)
(9)
And (8) written as
(10)
* *
1 2
2
2 (x , x )
/
/
f x
g x
λ
∂ ∂
= −
∂ ∂
* *
1 2
1 1 (x , x )
0
f g
x x
λ
⎛ ⎞∂ ∂
+ =⎜ ⎟
∂ ∂⎝ ⎠
* *
1 2
2 2 (x , x )
0
f g
x x
λ
⎛ ⎞∂ ∂
+ =⎜ ⎟
∂ ∂⎝ ⎠
10. D Nagesh Kumar, IISc Optimization Methods: M2L410
Solution by method of Lagrange
multipliers…contd.
Also, the constraint equation has to be satisfied at the extreme point
(11)
Hence equations (9) to (11) represent the necessary conditions for the
point [x1*, x2*] to be an extreme point.
Note that λ could be expressed in terms of as well and
has to be non-zero.
Thus, these necessary conditions require that at least one of the partial
derivatives of g(x1 , x2) be non-zero at an extreme point.
* *
1 2
1 2 ( , )
( , ) 0x x
g x x =
1/g x∂ ∂ 1/g x∂ ∂
11. D Nagesh Kumar, IISc Optimization Methods: M2L411
Solution by method of Lagrange
multipliers…contd.
The conditions given by equations (9) to (11) can also be generated by
constructing a functions L, known as the Lagrangian function, as
(12)
Alternatively, treating L as a function of x1,x2 and λ, the necessary
conditions for its extremum are given by
(13)
1 2 1 2 1 2( , , ) ( , ) ( , )L x x f x x g x xλ λ= +
1 2 1 2 1 2
1 1 1
1 2 1 2 1 2
2 2 2
( , , ) ( , ) ( , ) 0
( , , ) ( , ) ( , ) 0
( , , ) ( , ) 0
L f g
x x x x x x
x x x
L f g
x x x x x x
x x x
L
x x g x x
λ λ
λ λ
λ1 2 1 2
λ
∂ ∂ ∂
= + =
∂ ∂ ∂
∂ ∂ ∂
= + =
∂ ∂ ∂
∂
= =
∂
12. D Nagesh Kumar, IISc Optimization Methods: M2L412
Necessary conditions for a general
problem
For a general problem with n variables and m equality constraints the
problem is defined as shown earlier
Maximize (or minimize) f(X), subject to gj(X) = 0, j = 1, 2, … , m
where
In this case the Lagrange function, L, will have
one Lagrange multiplier λj for each constraint
as
(14)
1 2 , 1 2 1 1 2 2( , ,..., , ,..., ) ( ) ( ) ( ) ... ( )n m m mL x x x f g g gλ λ λ λ λ λ= + + + +X X X X
1
2
n
x
x
x
⎧ ⎫
⎪ ⎪
⎪ ⎪
= ⎨ ⎬
⎪ ⎪
⎪ ⎪⎩ ⎭
X
M
13. D Nagesh Kumar, IISc Optimization Methods: M2L413
Necessary conditions for a general
problem…contd.
L is now a function of n + m unknowns, , and the
necessary conditions for the problem defined above are given by
(15)
which represent n + m equations in terms of the n + m unknowns, xi and λj.
The solution to this set of equations gives us
(16)
and
1 2 , 1 2, ,..., , ,...,n mx x x λ λ λ
1
( ) ( ) 0, 1,2,..., 1,2,...,
( ) 0, 1,2,...,
m
j
j
ji i i
j
j
gL f
i n j m
x x x
L
g j m
λ
λ
=
∂∂ ∂
= + = = =
∂ ∂ ∂
∂
= = =
∂
∑X X
X
*
1
*
2
*
n
x
x
x
⎧ ⎫
⎪ ⎪
⎪ ⎪
= ⎨ ⎬
⎪ ⎪
⎪ ⎪⎩ ⎭
X
M
*
1
*
* 2
*
m
λ
λ
λ
λ
⎧ ⎫
⎪ ⎪
⎪ ⎪
= ⎨ ⎬
⎪ ⎪
⎪ ⎪⎩ ⎭
M
14. D Nagesh Kumar, IISc Optimization Methods: M2L414
Sufficient conditions for a general problem
A sufficient condition for f(X) to have a relative minimum at X* is that each
root of the polynomial in Є, defined by the following determinant equation
be positive.
(17)
11 12 1 11 21 1
21 22 2 12 22 2
1 2 1 2
11 12 1
21 22 2
1 2
0
0 0
0 0
n m
n m
n n nn n n mn
n
n
m m mn
L L L g g g
L L L g g g
L L L g g g
g g g
g g g
g g g
−∈
−∈
−∈
=
L L
M O M M O M
L L
L L L
M O M
M O M M M
L L L
15. D Nagesh Kumar, IISc Optimization Methods: M2L415
Sufficient conditions for a general
problem…contd.
where
(18)
Similarly, a sufficient condition for f(X) to have a relative maximum at X*
is that each root of the polynomial in Є, defined by equation (17) be
negative.
If equation (17), on solving yields roots, some of which are positive and
others negative, then the point X* is neither a maximum nor a minimum.
2
* *
*
( , ), for 1,2,..., 1,2,...,
( ), where 1,2,..., and 1,2,...,
ij
i j
p
pq
q
L
L i n and j m
x x
g
g p m q
x
λ
∂
= = =
∂ ∂
∂
= = =
∂
X
X n
16. D Nagesh Kumar, IISc Optimization Methods: M2L416
Example
Minimize , Subject to
or
2 2
1 1 2 2 1 2( ) 3 6 5 7 5f x x x x x x= − − − + +X 1 2 5x x+ =
17. 1 2 1
2
1 2 1
1 2 2 1
6 10 5 0
1
3 5 (5 )
2
1
3( ) 2 (5 )
2
x x
x
x x
x x x
λ
λ
λ
∂
= − − + + =
∂
=> + = +
=> + + = +
L
2
1
2
x
−
=
1
11
2
x =
[ ]
1 11
* , ; * 23
2 2
−⎡ ⎤
= =⎢ ⎥⎣ ⎦
X λ 11 12 11
21 22 21
11 12
0
0
L L g
L L g
g g
−∈⎛ ⎞
⎜ ⎟
−∈ =⎜ ⎟
⎜ ⎟
⎝ ⎠