Engineering Mathematics

A new nonlinear integral-equation model is derived in terms of hodograph variables for free-surface flow past an arbitrary bottom obstruction. A numerical method, carefully chosen to solve the resulting nonlinear algebraic equations and a simple, yet effective radiation condition have led to some very encouraging results. In this paper, results are presented for a semi-circular obstruction and are compared with

This paper discusses and evaluates strategies at the University of Sheffield for both engaging new engineering students in a proper appreciation of their mathematical standing and helping them develop the requisite skills. Within this context, the paper reviews the efficacy and role of a recent move by an engineering department to teach the mathematics module in-house.
The need to encourage students to reflect on which mathematical topics they may need to reinforce quickly and to give them proper direction on how to go about this is well understood. However, there is also a desire from parent departments to understand the needs of the whole cohort alongside the early identification of individuals who may need extra support. In 2009, for the first time, certainly at Sheffield, a diagnostic instrument was delivered and monitored by the Maths and Statistics Help Centre rather than each individual department doing its own thing. The advantage of such an approach was consistency across all engineering departments; secondary benefits included the involvement of both students and first year tutors in the support available centrally from the centre.

1. Let í µí°¼ = {0, ±1, ±2, ±3 … … … } and í µí± = {(í µí±Ž, í µí±): í µí±Ž−í µí± 4 = í µí°¾, í µí°¾ ∈ í µí°¼} is an equivalence relation then find the equivalence classes. 2. Show that the relation R on the set í µí°´=µí°´= {í µí±¥ ∈ í µí±: 0 ≤ í µí±¥ ≤ 12} given by í µí± = {(í µí±Ž, í µí±): |í µí±Ž − í µí±| í µí±–í µí± í µí±ší µí±¢í µí±™í µí±¡í µí±–í µí±í µí±™í µí±’ í µí±œí µí±“ 4} is an equivalence relation. Also find the equivalence class [1]. 3. Prove that the relation R on the set í µí± × í µí± defind by (í µí±Ž, í µí±)í µí±(í µí±, í µí±‘) ⇔ í µí±Ž + í µí±‘ = í µí± + í µí± ∀ (í µí±Ž, í µí±), (í µí±, í µí±‘) ∈ í µí± × í µí± is equivalence relation. Find the equivalence class. Question based on Domain and Range of the function 4. Let í µí±“: í µí± → í µí±: í µí±“(í µí±¥) = 2í µí±¥−1 2 and í µí±”: í µí±„ → í µí± í µí±”(í µí±¥) = í µí±¥ + 2 be two functions then (í µí±”í µí±œí µí±“) (3 2). 5. Let í µí±“(í µí±¥) = í µí±¥ 2 − í µí±¥ + 1, í µí±¥ ≥ 1 2 , then the solution of the equation í µí±“(í µí±¥) = í µí±“ −1 (í µí±¥). 6. If í µí±“(í µí±¥) = í µí±¥ í µí±¥−1 , í µí±¥ ≠ 1 then find (í µí±“í µí±œí µí±“í µí±œí µí±“í µí±œí µí±“ … … … …. . í µí±œí µí±“)(í µí±¥) {f is 19 times}. 7. Find the range of the function í µí±“(í µí±¥) = í µí±¥ 2 +í µí±¥+2 í µí±¥ 2 +í µí±¥+1 , í µí±¥ ∈ í µí±. 8. Find the range of í µí±¦ = [2í µí±¥]. 9. Find the domain of the function í µí°¹(í µí±¥) = 1 [í µí±¥] + √2í µí±¥ − í µí±¥ 2. 10. Find the range of the function í µí±“(í µí±¥) = cos í µí±¥ − sin í µí±¥.

Abstract
Paper hypothesis. 
- There's a light beam its velocity (1.16 mkm/) moves in parallel to Mercury Motion.

Paper Hypotheses Explanation
- Planet Motion be a cover for a light motion found behind. The light motion uses Planet motion data for its own motion.
- The hypothesis creates one more universe in parallel to the universe of matter. And by that, The Planets be similar to ships sail over a sea. But the sea is light motions network.
- The hypothesis uses Mercury Motion as a examination point to prove that a motion of a light beam its velocity (1.16 mkm/s) be behind Mercury Motion. 
- The light motion causes to create the solar system distances in a network form and depends on one geometrical design. Because of that, the solar system distances analysis will prove the light supposed velocity (1.16 mkm/s) existence.
Paper Conclusion
- The Solar System Distances Be Created in a network form and depend on one geometrical design which proves that these distances be created by a motion of a light beam its velocity =1.16 mkm/sec.
Paper Objective
- The paper proves The Light Supposed Velocity (1.16 mkm/sec) Existence.
- This proof disproves automatically Newton Theory Of The Sun Mass Gravity. 
- Because
- The sun rays velocity = 300000 km/sec.
- Where
- The solar system distances prove that they are created by a light beam its velocity =1.16 mkm/sec.
- means,
- The sun light velocity (300000 km/) is created as a side product from the light supposed velocity (1.16 mkm/s),
- means
- The Sun Is Created After All Solar Planets Creation And Motion
- Which disproves Decisively Newton Theory Of The Sun Mass Gravity.
Gerges Francis Tawdrous +201022532292
Physics Department-  Physics & Mathematics  Faculty

Introduction to s-plane (complex frequency plane) for engineering circuit analysis.

s-plane study is the 's' in electric circuit problem solving applying Laplace method.

Engineering Electrical Courses Mostly Used In:
1. upper level electric circuits
2. electrical control systems
3. signals and systems
4. signal processing
4. filters . 

Study here is building up from basics.
What is the s-plane type of study.
Level: Intermediate for electrical circuits course.

Reference: Hyat and Kemmerly, Engineering Circuit Analysis 4th edition.

Apologies for any errors and omissions.

Probability theory is a fundamental tool of many disciplines of science and engineering. It is a mathematical framework for representing uncertain statements. It provides the means of quantifying uncertainty as well as axioms for deriving new uncertain statements. In articial intelligence applications, we use probability theory in two major ways. First, the laws of probability tell us how AI systems should reason, so we design our algorithms to compute or approximate various expressions derived using probability theory. Second, we can use probability and statistics to theoretically analyze the behavior of proposed AI systems. I. WHY PROBABILITY? There are three possible sources of uncertainty: • Inherent stochasticity in the system being modeled. For example, most interpretations of quantum mechanics describe the dynamics of subatomic particles as being probabilistic. We can also create theoretical scenarios that we postulate to have random dynamics, such as a hypothetical card game where we assume that the cards are truly shuffled into a random order. • Incomplete observability. Even deterministic systems can appear stochastic when we cannot observe all the variables that drive the behavior of the system. For example, in the Monty Hall problem, a game show contestant is asked to choose between three doors and wins a prize held behind the chosen door. Two doors lead to a goat while a third leads to a car. The outcome given the contestant's choice is deterministic, but from the contestant's point of view, the outcome is uncertain. • Incomplete modeling. When we use a model that must discard some of the information we have observed, the discarded information results in uncertainty in the model's predictions. For example, suppose we build a robot that can exactly observe the location of every object around it. If the robot discretizes space when predicting the future location of these objects, then the discretization makes the robot immediately become uncertain about the precise position of objects: each object could be anywhere within the discrete cell that it was observed to occupy. In many cases, it is more practical to use a simple but uncertain rule rather than a complex but certain one, even if the true rule is deterministic and our modeling system has the fidelity to accommodate a complex rule. For example, the simple rule " Most birds fly " is cheap to develop and is broadly useful, while a rule of the form, " Birds y, except for very young birds that have not yet learned to y, sick or injured birds that have lost the ability to y, ightless species of birds including the cassowary, ostrich and kiwi.. . " is expensive to develop, maintain and communicate and, after all this effort, is still brittle and prone to failure. Probability theory was originally developed to analyze the frequencies of events. It is easy to see how probability theory can be used to study events like drawing a certain hand of cards in a poker game. These kinds of events are often repeatable. When we say that an outcome has a probability p of occurring, it means that if we repeated the experiment (e.g., drawing a hand of cards) infinitely many times, then proportion p of the repetitions would result in that outcome. Probability can be seen as the extension of logic to deal with uncertainty. Logic provides a set of formal rules for determining what propositions are implied to be true or false given the assumption that some other set of propositions is true or false. Probability theory provides a set of formal rules for determining the likelihood of a proposition being true given the likelihood of other propositions. II. INFORMATION THEORY Information theory is a branch of applied mathematics that revolves around quantifying how much information is present in a signal. It was originally invented to study sending messages from discrete alphabets over a noisy channel, such as communication via radio transmission. In this context, information theory tells how to design optimal codes and calculate the expected length of messages sampled from specic

In this paper we mention cut vertex and cut edge in a connected grapg. We establish a minimally connected graph with no cycles. And then, a graph G with n vertices, n-1 edges and no cycles, it is connected. Finally, G contains trees, whose minimum degree, δ(G) ≥ k  and it is shown that the ordre of subgraph tree with at most δ(G)+1.

This study was conducted to determine the effectiveness of peer tutoring on the mathematics performance of fourth year slow-learners. The specific questions answered were: 1. What are the pretest and posttest mean scores of the control group? 2. What are the pretest and posttest mean scores of the experimental group exposed to peer tutoring program? 3. Is there a significant difference between the pretest and posttest mean scores of the control group? 4. Is there a significant difference between the pretest and posttest mean scores of the experimental group exposed to peer tutoring? 5. Is there a significant difference in the mean gain scores between the control group and the experimental group? This study made use of modified experimental design 10, otherwise known as the non-equivalent control group design. It involved one hundred twenty two (122) fourth year high school students of Daniel R. Aguinaldo National High School at Matina, Davao City for the School Year 2003-2004. Out of these numbers, sixty (60) students served as the experimental subjects and sixty two (62) students from other class served as the control subjects. Based on the findings of this study, the following conclusions are drawn: 1. The use of remedial instruction indicates an increase in the student's achievement. 2. The Peer Tutoring Approach as remedial instruction in the Experimental Group showed that there is no difference in the achievement result in the Control Group. The researcher offered the following recommendations: 1. Teachers who do remediation activities in any subject may try Peer Tutoring as an alternative approach with the same outcomes as the traditional instruction approach. 2. School Administrators like the Principals and Head Teachers should allocate at least 30 minutes, three times a week meetings for remedial instruction.

The Pythagorean theorem is perhaps the best known theorem in the vast world of mathematics. A simple relation of square numbers, which encapsulates all the glory of mathematical science, is also justifiably the most popular yet sublime theorem in mathematical science. The starting point was Diophantus' 20 th problem (Book VI of Diophantus' Arithmetica), which for Fermat is for n = 4 and consists in the question whether there are right triangles whose sides can be measured as integers and whose surface can be square. This problem was solved negatively by Fermat in the 17 th century, who used the wonderful method (ipse dixit Fermat) of infinite descent. The difficulty of solving Fermat's equation was first circumvented by Willes and R. Taylor in late 1994 ([1],[2],[3],[4]) and published in Taylor and Willes (1995) and Willes (1995). We present the proof of Fermat's last theorem and other accompanying theorems in 4 different independent ways. For each of the methods we consider, we use the Pythagorean theorem as a basic principle and also the fact that the proof of the first degree Pythagorean triad is absolutely elementary and useful. The proof of Fermat's last theorem marks the end of a mathematical era; however, the urgent need for a more educational proof seems to be necessary for undergraduates and students in general. Euler's method and Willes' proof is still a method that does not exclude other equivalent methods. The principle, of course, is the Pythagorean theorem and the Pythagorean triads, which form the basis of all proofs and are also the main way of proving the Pythagorean theorem in an understandable way. Other forms of proofs we will do will show the dependence of the variables on each other. For a proof of Fermat's theorem without the dependence of the variables cannot be correct and will therefore give undefined and inconclusive results.

This is an unwelcomed but hard-hitting truth about the issues within Mathematics. This question kept bugging me after I found the error, and after evaluation, there was no possible way for NASA to know if the O-RINGS were damaged after the explosion. Tests cannot reproduce. There is no possible way for the ORINGS to have been frozen given the situation of atmospheric temperature and the radiation of heat from the rockets. It was not a cover-up. I truly believe they did not understand why. The public would not be kind to "we do not know."

The stability of steel beams with discrete lateral restraints is analysed using the Rayleigh–Ritz method, with the degrees-of-freedom being represented by Fourier series. Linear eigenvalue analysis of the system shows that the system buckles into one of two classes of mode, either node-displacing or between bracing nodes. Buckling mode progression behaviour is examined and limiting characteristics of the system are found. Comparison is made with a single harmonic representation of the DOFs, which is found to lead to erroneous predictions. Validation of the analysis is performed using the finite element method and numerical continuation. Finally, it is shown that design formulae based on analogous column behaviour can return both conservative and unsafe results, depending on brace location.

We present a number of exact solutions to the linearised Grad equations for non-equilibrium rarefied gas flows and heat flows. The solutions include the flow and pressure fields associated to a point force placed in a rarefied gas flow close to a no-slip boundary and the temperature field for a point heat source placed in a heat flow close to a temperature jump boundary. We also derive the solution of the unsteady Grad equations in one dimension with a time-dependent point heat source term and the Grad analogue of the rotlet, a well-known singularity of Stokes flow which corresponds to a point torque.

In this paper, we describe two different finite difference schemes for solving the time fractional diffusion equation. And, we study the method of lines discretizations. Then, we use to check the stability of finite difference schemes by using Von Neumann analysis.