This document contains the details of an end-semester examination for an undergraduate course on Theory of Computation. The exam consists of 10 multiple choice and justification questions (Part 1) and 2 longer proof questions (Parts 2 and 3). Part 1 asks students to determine whether statements about formal language classes are true or false and provide justifications. Part 2 asks students to prove that a particular language defined using pushdown automata is not context-free. Part 3 asks students to design an algorithm to show that a language involving matching strings is recursive.
After a long period, I bring you new - fresh Presentation which gives you a brief idea on sub-problem of Dynamic Programming which is called as -"Longest Common Subsequence".I hope this presentation may help to all my viewers....
Nodal Domain Theorem for the p-Laplacian on Graphs and the Related Multiway C...Francesco Tudisco
We consider the p-Laplacian on discrete graphs, a nonlinear operator that generalizes the standard graph Laplacian (obtained for p=2). We consider a set of variational eigenvalues of this operator and analyze the nodal domain count of the corresponding eigenfunctions. In particular, we show that the famous Courant’s nodal domain theorem for the linear Laplacian carries over almost unchanged to the nonlinear case. Moreover, we use the nodal domains to prove a higher-order Cheeger inequality that relates the k-way graph cut to the k-th variational eigenvalue of the p-Laplacian
This PPT will clarify your all doubts in Arithmetic Progression.
Please download this PPT and if any doubt according to this PPT, please comment , then i will try to solve your problem.
Thank you :)
After a long period, I bring you new - fresh Presentation which gives you a brief idea on sub-problem of Dynamic Programming which is called as -"Longest Common Subsequence".I hope this presentation may help to all my viewers....
Nodal Domain Theorem for the p-Laplacian on Graphs and the Related Multiway C...Francesco Tudisco
We consider the p-Laplacian on discrete graphs, a nonlinear operator that generalizes the standard graph Laplacian (obtained for p=2). We consider a set of variational eigenvalues of this operator and analyze the nodal domain count of the corresponding eigenfunctions. In particular, we show that the famous Courant’s nodal domain theorem for the linear Laplacian carries over almost unchanged to the nonlinear case. Moreover, we use the nodal domains to prove a higher-order Cheeger inequality that relates the k-way graph cut to the k-th variational eigenvalue of the p-Laplacian
This PPT will clarify your all doubts in Arithmetic Progression.
Please download this PPT and if any doubt according to this PPT, please comment , then i will try to solve your problem.
Thank you :)
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Solutions Manual for An Introduction To Abstract Algebra With Notes To The Fu...Aladdinew
Full download : https://goo.gl/XTCXti Solutions Manual for An Introduction To Abstract Algebra With Notes To The Future Teacher 1st Edition by Nicodemi
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.
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
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The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
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.
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Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
End semexam | Theory of Computation | Akash Anand | MTH 401A | IIT Kanpur
1. MTH 401: Theory of Computation April 22, 2014
Department of Mathematics and Statistics Time: 180 minutes
Indian Institute of Technology - Kanpur Maximum Score: 40
End-semester Examination
1. Indicate whether following statements are true or false. Justify your answer (to justify
a claim that a statement is true, an (informal) proof is required; to justify a claim
that a statement is false, a single counterexample is sufficient.) Note that no credit
will be given to a correct guess without any explanation or followed by an
incorrect justification.
(a) The language {0, 1}∗
is countable. [1]
True. Clearly,
{0, 1}∗
=
∞
n=0
{w ∈ {0, 1}∗
: |w| = n},
is a countable union of finite sets, and therefore, countable.
(b) The regular expression 0∗
(100 + 010 + 001)0∗
generates the language {w | w ∈
{0, 1}∗
and w contains at least two 0s and at most one 1}. [1]
False. Clearly, a string generated by the given regular expression always contains
exactly one 1.
(c) If a language is accepted by a non-deterministic finite state machine, then it is
clearly context-free. [1]
True. A language that is accepted by a non-deterministic FSM is regular, and
all regular languages are context-free.
(d) If L1L2 is regular then L1 and L2 are both regular. [1]
False. L1 = L(0∗
) and L2 = {0p
| p is a prome}. Clearly, L1L2 = L(0∗
){ , 0, 00}
is regular whereas L2 is not regular.
(e) If L∗
= ∅, then L = ∅. [1]
False. An equivalent statement is: if L = ∅, then L∗
= ∅. Clearly false, as ∈ L∗
even when L = ∅.
(f) If a and b are letters in an alphabet, then (a∗
b∗
)∗
= (a + b)∗
. [1]
True. w ∈ L((a∗
b∗
)∗
) ⇐⇒ there are integers m1, . . . , mk ≥ 0, and n1, . . . , nk ≥
0 for some k ≥ 0 such that w = an1
bm1
. . . ank bmk ⇐⇒ w ∈ L((a + b)∗
).
(g) The language given by {w ∈ {0, 1}∗
| n0(w) = n1(w)}, where na(w) denotes the
number of occurrences of the symbol a in w, is a context free language. [1]
True. {w ∈ {0, 1}∗
| n0(w) = 2n1(w)} = L(G) where G = ({0, 1}, {S}, S, {S →
0S0S1 | 0S1S0 | 1S0S0}).
2. 2
(h) Let L be a language over the alphabet {0, 1}. If L∗
is regular, then L is also
regular. [1]
False. Let L = {0, 1} ∪ {0n
1n
| n ≥ 0}. Then L∗
= {0, 1}∗
is clearly regular
where as L is not regular.
(i) The set of Turing machines over a given alphabet that do not accept any even
length palindrome is decidable. [1]
False. By Rice’s Theorem.
(j) If L1 is Recursive and L2 is PSPACE-complete then there is a reduction from
L1 to L2. [1]
False. For example, any recursive language in EXP that is not in PSPACE can
not be reduced to a PSAPCE-complete language.
2. Let P be the set of all pushdown machines and e : P → {0, 1}∗
be a given function.
Let L = {w ∈ Σ∗
| w ∈ L(w)}, where L(w) = L(P) if w = e(P). Note that L(w) = ∅
if w ∈ e(P). Show that L is not a context free language. [10]
Solution: Suppose L is context free. Then, there is a pushdown machine P0 such that
L = L(P0). Let w0 = e(P0). If w0 ∈ L = L(P0) = L(w0), then, by definition of L,
w0 ∈ L. A contradiction. Similarly, if w0 ∈ L = L(P0) = L(w0), then, by definition of
L, w0 ∈ L. Again a contradiction. Thus, L is not context free.
3. Let Σ = {a}. Suppose L0 = {(x1, y1), (x2, y2), . . . , (xn, yn) | xi, yi ∈ Σ∗
} be a language
over the alphabet Σ0 = Σ ∪ {(} ∪ {)} ∪ {, } and
LPCP = {(x1, y1), . . . , (xn, yn) ∈ L0 |
there exists (i1, . . . , im) with 0 ≤ il ≤ n such that xi1 · · · xim = yi1 · · · yim }.
Show that LPCP is recursive. [10]
Solution: In order to show that LPCP is recursive, we need to provide a membership
algorithm. Note that every p = (x1, y1), (x2, y2), . . . , (xn, yn) ∈ L0 is of the form
p = (al1
, am1
), (al2
, am2
), . . . , (aln
, amn
) for some l1, l2, . . . , ln and m1, m2, . . . , mn.
Thus, given any p = (al1
, am1
), (al2
, am2
), . . . , (aln
, amn
) ∈ L0, the following algorithm
decides its membership to LPCP :
Step 1: For each i = 1 to n, compute di = li − mi.
Step 2: If for all i0 ∈ {1, 2, . . . , n}, di > 0, then p ∈ LPCP .
Step 3: Else if for all i0 ∈ {1, 2, . . . , n}, di < 0, then p ∈ LPCP .
Step 4: Else if the is an i0 ∈ {1, 2, . . . , n} such that di0 = 0, then p ∈ LPCP as xi0 = yi0 .
3. 3
Step 5: Else there is an i1 ∈ {1, 2, . . . , n} such that di1 > 0 and i2 ∈ {1, 2, . . . , n} such
that di2 < 0. Then, p ∈ LPCP as xi1 . . . xi1
(−di2
) times
xi2 . . . xi2
(di1
) times
= a−di2
li1
+di1
li2 = a−di2
(di1
+mi1
)+di1
(di2
+mi2
)
=
a−di2
mi1
+di1
mi2 = yi1 . . . yi1
(−di2
) times
yi2 . . . yi2
(di1
) times
.
4. Recall that, given two languages L1 and L2, their right quotient, denoted by L1/L2,
is defined as {w ∈ Σ∗
: ∃x ∈ L2 such that wx ∈ L1}. Suppose Σ = {0, 1, 2, 3}. Let
L1 = L(G1) and L2 = L(G2) where
G1 = (Σ, {S}, S, {S → 12
30 | 03
31 | 01232 | 12
S0 | 03
S1 | 012S2})
and
G2 = (Σ, {S}, S, {S → 030 | 131 | 232 | 0S0 | 1S1 | 2S2}).
Show that L1/L2 is not a context free language. [10]
Hint: Observe what elements can belong to the set L1/L2 to argue why it can not be
context free.
Solution: Let L = L1/L2. Each string z in L is obtained from words z1 in L1 and z2
in L2 satisfying z1 = zz2. Since each string in L1 or L2 contains the symbol 3 exactly
once, the terminal substrings starting from 3 in z1 and in z2 are the same. If the string
30 (or 31) is a substring of z1, then 12
30 (or 03
31) occurs in z1 and 030 (or 131) occurs
in z2. Either case contradicts the equation z1 = zz2. Thus the only letter which can
occur immediately to the right of 3 in z1 is 2. Therefore z1 must contain 01232 as a
substring and z2 must contain 232 as a substring. Hence the shortest strings which can
occur as z1, z2 are 01232 and 232. Thus 01 is in L. In z1 we see that 232 is preceded
by 1. Then any longer word for z2 must contain 12321, and the corresponding z1
must contain 03
012321. Therefore 04
is in L. This line of reasoning can be continued
inductively to provide a means of enumerating all the elements of L. We find that L
consists of the sequence
01, 04
, 12
03
, 14
02
, 16
0, 18
, 03
17
, 06
16
, . . . , 024
, . . .
where to pass from one string xi in the sequence to the next xi+1, we use the following
rule:
• If xi = yi0, then xi+1 = 12
yi.
• If xi = yi1, then xi+1 = 03
yi.
In particular, 0n
∈ L if and only if n = 4(6)k
for k ≥ 0.
Now, if L is a CFL, then there exists a pumping lemma constant N. Choose a k0
large enough such that n0 = 4(6)k0
> N. Clearly z = 0n0
∈ L with |z| > N. Now,
irrespective of how we split z into five pieces, u, v, w, x, y with x = uvwxy, |uwy| ≤ N,
|vx| = l > 0, uvi
wxi
y = 0n0+(i−1)l
∈ L for some i ≥ 0, a contradiction. Thus, L is not
a CFL.