#Properties of Laplace transformation(Linearity

[yadavkhushboo]

ISRO Scientist Syllabus

The Indian Space Research Organisation (ISRO) is one of the leading space agencies globally and offers a wide range of opportunities for candidates aspiring to work as scientists and engineers. The ISRO Scientist syllabus is designed to assess candidates on their technical knowledge, problem-solving abilities, and aptitude in relevant engineering disciplines. Typically, the ISRO recruitment exam is conducted for various technical positions like Scientist/Engineer in different engineering fields, including Electronics, Mechanical, and Computer Science. Below is a breakdown of the typical syllabus for ISRO Scientist recruitment.

1. General Aptitude:

The General Aptitude section is designed to assess the candidate's logical reasoning, quantitative aptitude, and analytical skills. This section often includes topics like:

Quantitative Aptitude: Number systems, ratios and proportions, profit and loss, time and work, time and distance, simple and compound interest, percentages, average, and permutation-combination.

Logical Reasoning: Blood relations, series, direction sense, coding-decoding, seating arrangements, puzzles, and analogies.

Verbal Ability: Vocabulary, reading comprehension, sentence correction, and grammar.

This section tests the candidate's ability to think logically and solve problems efficiently within a short span of time.

2. Core Subject Knowledge:

The core subject knowledge tests are designed according to the specific engineering discipline of the candidate. For example:

Electronics Engineering (EC):

Circuit Theory: Network theorems, AC and DC circuits, transient analysis, resonance, and filters.

Digital Electronics: Logic gates, combinational and sequential circuits, flip-flops, counters, and registers.

Microprocessors and Microcontrollers: 8085/8086 microprocessors, interfacing, and assembly language programming.

Control Systems: Stability analysis, Bode plot, Nyquist plot, transfer functions, and feedback control systems.

Signals and Systems: Fourier series, Laplace transform, and Z-transform.

Communication Systems: Analog and digital communication, modulation techniques, and transmission systems.

Mechanical Engineering (ME):

Engineering Mechanics: Laws of motion, friction, dynamics, kinematics, and work-energy principles.

Thermodynamics: Laws of thermodynamics, entropy, heat engines, and refrigeration cycles.

Strength of Materials: Stress, strain, shear force, bending moment, and material properties.

Fluid Mechanics: Fluid statics, fluid dynamics, Bernoulli’s equation, and flow measurement.

Manufacturing Processes: Casting, welding, forming processes, and machining operations.

Computer Science Engineering (CS):

Data Structures and Algorithms: Arrays, linked lists, stacks, queues, trees, graphs, sorting, and searching algorithms.

Operating Systems: Process management, memory management, file systems, and system calls.

Databases: Relational databases, SQL, normalization, and transaction management.

Computer Networks: OSI model, TCP/IP, routing, and protocols.

Software Engineering: Software development lifecycle, methodologies, and design principles.

These core subjects assess the candidate's depth of understanding of the concepts and their application in real-world scenarios.

3. Technical Aptitude:

Technical aptitude involves the ability to apply theoretical concepts to solve practical engineering problems. This section may include topics such as:

Electromagnetics: Electric fields, magnetic fields, Maxwell's equations, and wave propagation.

Linear Algebra: Matrices, eigenvalues, eigenvectors, and systems of linear equations.

Material Science: Properties of materials, alloys, composites, and failure analysis.

Instrumentation: Sensors, actuators, and measurement techniques.

4. Exam Pattern:

The ISRO Scientist exam typically consists of multiple-choice questions (MCQs) with four options, and candidates must select the correct answer. The exam duration is usually two to three hours, and the total marks vary depending on the specific exam.

5. Preparation Strategy:

Know the Syllabus: Candidates must understand the syllabus thoroughly and focus on topics with high weightage.

Study Materials: Referring to standard textbooks, previous years' question papers, and mock tests will enhance the preparation process.

Time Management: Efficient time management is crucial while preparing and attempting the exam to ensure each section is covered adequately.

In conclusion, preparing for the ISRO Scientist exam requires a comprehensive understanding of core engineering subjects, aptitude, and problem-solving skills. Candidates need to stay focused, practice regularly, and follow a structured study plan to succeed in this competitive exam.

[iit-jam]

IIT JAM Syllabus 2025: A Comprehensive Guide

The IIT JAM (Joint Admission Test for Masters) is one of the most competitive exams for students aspiring to pursue postgraduate studies in esteemed institutions like IITs and IISc. Mathematics, being a core subject, attracts candidates with strong analytical and problem-solving skills. To excel in this exam, a thorough understanding of the IIT JAM Mathematics Syllabus 2025 is essential. This blog outlines the syllabus in detail and provides tips to help candidates prepare effectively.

Overview of IIT JAM Mathematics Syllabus 2025

The IIT JAM Mathematics Syllabus 2025 is crafted to test the candidates' knowledge of fundamental mathematical concepts covered at the undergraduate level. The syllabus is broad, covering topics such as calculus, linear algebra, differential equations, and numerical analysis. Each section focuses on key areas that are crucial for advanced studies and professional applications.

Key Topics in the Syllabus

1. Sequences and Series

This section includes the convergence of sequences and series, tests for convergence (such as comparison, ratio, and root tests), and the study of power series and their radius of convergence.

2. Differential Calculus

Candidates must understand single-variable calculus concepts like limits, continuity, and differentiability. Topics also include Taylor series, mean value theorem, and indeterminate forms. For multivariable calculus, partial derivatives, maxima, minima, saddle points, and the method of Lagrange multipliers are essential.

3. Integral Calculus

This section covers definite and indefinite integrals, improper integrals, and special functions like beta and gamma functions. The application of double and triple integrals is also emphasized.

4. Linear Algebra

A critical area of the syllabus, it focuses on vector spaces, subspaces, linear transformations, rank, nullity, eigenvalues, eigenvectors, and matrix diagonalization. Understanding the solution of systems of linear equations is vital.

5. Real Analysis

This section involves the properties of real numbers, limits, continuity, differentiability, and Riemann integration. Candidates must also be familiar with sequences, Cauchy sequences, and uniform continuity.

6. Ordinary Differential Equations (ODEs)

This includes first-order ODEs, linear differential equations with constant coefficients, systems of linear ODEs, and Laplace transform techniques for solutions.

7. Vector Calculus

Important topics include gradient, divergence, curl, line integrals, surface integrals, and volume integrals, along with Green’s, Stokes’, and Gauss divergence theorems.

8. Group Theory

The basics of groups, subgroups, cyclic groups, Lagrange’s theorem, permutation groups, and homomorphisms are covered.

9. Numerical Analysis

This section focuses on numerical solutions for non-linear equations, numerical integration and differentiation, interpolation methods, and error analysis.

Tips for Preparing the Syllabus

Understand the Weightage: Review past papers to prioritize high-scoring topics like Linear Algebra, Real Analysis, and Differential Calculus.

Strategize Your Study Plan: Divide the syllabus into manageable sections, set achievable goals, and stick to a consistent schedule.

Practice Regularly: Solve previous years’ papers and mock tests to familiarize yourself with the question patterns and improve speed.

Strengthen Fundamentals: Focus on core concepts by revisiting undergraduate textbooks and seeking clarity on challenging topics.

Leverage Online Resources: Utilize tutorials, study materials, and practice tests available online to supplement your preparation.

Conclusion

The IIT JAM Mathematics Syllabus 2025 is extensive yet well-structured, providing a clear framework for aspirants to plan their preparation. By mastering the syllabus and practicing diligently, candidates can confidently tackle the exam and achieve their dream of joining top postgraduate programs. Dedicate time, stay consistent, and focus on strengthening your mathematical foundations to excel in IIT JAM Mathematics 2025.

[erikabsworld]

Mastering Linear System Modeling: A Step-by-Step Guide

Are you struggling with linear system modeling assignments? Do you find yourself lost in a maze of equations and concepts? Fear not! In this comprehensive guide, we'll demystify linear system modeling and provide you with a step-by-step approach to tackle even the toughest assignment questions. Let's dive in!

Understanding Linear System Modeling

Linear system modeling is a fundamental concept in various fields such as engineering, physics, economics, and more. It involves representing real-world systems using mathematical models to analyze their behavior and predict outcomes. These systems can range from electrical circuits to chemical processes to economic markets.

At its core, linear system modeling deals with relationships between inputs and outputs of a system, assuming linearity, which means the output is directly proportional to the input. This simplifies the analysis and allows for the use of powerful mathematical tools like matrix algebra and differential equations.

Sample Assignment Question:

Consider a spring-mass-damper system with the following properties:

Mass (m) = 1 kg

Spring constant (k) = 10 N/m

Damping coefficient (c) = 2 Ns/m

Given an external force (F(t)) of 5 N acting on the system, find the equation of motion and determine the system's response.

Step-by-Step Guide:

Understand the System: Before diving into calculations, it's crucial to understand the system's components and behavior. In this case, we have a spring (providing restorative force), a mass (subject to inertia), and a damper (dissipating energy).

Formulate the Equation of Motion: The equation of motion for this system can be derived using Newton's second law, which states that the sum of forces acting on an object equals its mass times acceleration (F = ma). In our case, the equation is: m(d^2x/dt^2) + c(dx/dt) + kx = F(t) . Where x represents the displacement of the mass from its equilibrium position.

Solve the Differential Equation: This is where we apply mathematical techniques to solve the differential equation obtained in the previous step. Depending on the nature of the external force (F(t)), solutions can vary. Common methods include Laplace transforms, numerical integration, or analytical solutions for simpler cases.

Analyze the System's Response: Once we have the solution for x(t), we can analyze the system's behavior over time. This includes studying transient and steady-state responses, stability, frequency response, and any other relevant characteristics.

How We Can Help:

At matlabassignmentexperts.com, we understand the challenges students face when dealing with complex topics like linear system modeling. That's why we offer expert linear system modeling assignment help to guide you through your assignments, ensuring clarity and accuracy in your solutions. Our team of experienced tutors is dedicated to providing personalized support tailored to your specific needs, helping you excel in your studies and achieve academic success.

In conclusion, mastering linear system modeling requires a solid understanding of the underlying principles and a systematic approach to problem-solving. By following the steps outlined in this guide and seeking assistance when needed, you can confidently tackle any assignment question that comes your way. Remember, practice makes perfect, so don't hesitate to put your newfound knowledge to the test!

[yokowan]

laplace transforms are a way of cheating at solving differential equations of one variable by turning them into algebraic equations of a different variable.

you just plug both sides of the equation into this weird looking integral:

since you're integrating with respect to t, the resulting transformed equation is no longer a function of t, it's a function of the new variable introduced, s.

on the surface this seems totally useless: you have to do a bunch of annoying calculus to turn one equation you don't know how to solve into another equation that's in terms of variables you don't care about. why would you do this? well, there's two useful properties.

first, the Laplace transforms of common functions are already known and they combine together in predictable ways. you never actually have to calculate them from the integral definition, you look up in a table what the transforms of different parts of the function are and just add them together. it's just like how you never solve integrals using the limit definition, you know the integrals of a few common functions and how they combine.

but Laplace transforms being easy to compute is one thing, it doesn't make them useful. what makes them useful is the way they work with derivatives. let's say our dependant variable x(t) has the Laplace transform X(s). then, if you take Laplace transform of the derivative x'(t), it works out to be sX(s) + x(0).

the Laplace transform of any nth derivative of a function is equal to the transform of the original function times s^n plus the initial condition. so, given the differential equation

x''(t) + 3x'(t) - 2x(t) = 0

the Laplace transform becomes

s²X(s) + 3sX(s) - 2X(s) - x'(0) + 3x(0) = 0

this equation is totally algebraic! you only have first order terms of X(s) with no derivatives! and then when you solve for X(s), you can use the same table you started with but in reverse to calculate the *inverse* Laplace transform, and you get the original function x(t). you just solved the differential equation doing absolutely no calculus! it's magical! for simple, linear ODEs it's definitely easier to just solve them the normal way, but with a lot of complicated non-linear differential equations, the Laplace transform makes unsolvable looking problems trivial!

if you didn't want an explanation and are annoyed that I bothered then uhhhhh pretend this isn't addressed to you and is instead addressed to any of my followers that currently might be studying diffeq kthxbye

toxic yuri relationship with differential equations

[turitolive]

Congruence Transformation  

Transformation   

A transformation is a function that maps points or objects from one space to another. In geometry, a transformation can be thought of as changing an object's position, orientation, or size.  

There are many diverse transformations in mathematics, including translation, rotation, scaling, reflection, shearing, and dilation. Moreover, these transformations can be applied to objects in different spaces, such as points in a two-dimensional or three-dimensional space, functions, or vectors in a linear space.  

Transformations are used in various mathematical applications, including geometry, calculus, linear algebra, and differential equations. They are particularly important in the study of symmetry and invariance, where they are used to analyze the properties of objects that remain unchanged under certain transformations.  

In addition to geometric transformations, there are many other types of transformations in mathematics, such as Fourier transforms, Laplace transforms, and wavelet transforms, which are used in signal processing, image processing, and other areas of applied mathematics.

Overall, transformations play a fundamental role in many areas of mathematics and are essential tools for modeling and analyzing mathematical systems.  

Types of Transformations  

In mathematics, several types of transformations are commonly used. Some of the most common types of transformations include:  

Translation: This transformation involves moving an object from one location to another. In a two-dimensional plane, translation involves moving an object horizontally and/or vertically.  

Rotation: This transformation involves turning an object around a fixed point. In a two-dimensional plane, rotation involves rotating an object by a certain angle around a point.  

Reflection: This transformation involves creating a mirror image of an object. In a two-dimensional plane, reflection involves flipping an object across a line (such as the x-axis or y-axis).  

Dilation: This transformation involves changing the size of an object relative to a fixed point. In a two-dimensional plane, dilation involves stretching or shrinking an object from a fixed point.  

What is a Congruence Transformation?  

A congruence transformation is a type of transformation in mathematics that preserves the size and shape of an object. For example, in linear algebra, a congruence transformation refers to a linear transformation that preserves the dot product of vectors.  

More specifically, let A and B be two n x n matrices. We say that A and B are congruent if there exists an invertible n x n matrix P such that:  

B = P^TAP  

where P^T is the transpose of P.  

Geometrically, this means that if we have a matrix A that represents a linear transformation, and we apply a congruence transformation to it by multiplying it on both sides by an invertible matrix P, the resulting matrix B represents the same linear transformation, but with a different coordinate system.  

Congruence transformations are useful in many areas of mathematics, including linear algebra, geometry, and number theory. For example, they can simplify matrix calculations, diagonalize symmetric matrices, and classify quadratic forms.  

Applications of Congruence Transformation  

Congruence transformations have many applications in mathematics and beyond. Here are some examples:  

Geometry: Congruence transformations are fundamental concepts in geometry, and they are used to study the properties of geometric objects such as points, lines, circles, and polygons. They are used to prove geometric theorems and to develop geometric models of physical phenomena.  

Computer graphics: Congruence transformations are used extensively to model and manipulate 2D and 3D objects. They are used to perform operations such as rotation, translation, scaling, and reflection, which are essential for creating realistic animations and visual effects.  

Physics: Congruence transformations are used in physics to study the properties of invariant physical systems under certain transformations.  

Robotics: Congruence transformations are used in robotics to model the movements of robots and to control their motion. 

Molecular biology: Congruence transformations are used in molecular biology to study the three-dimensional structure of biomolecules such as proteins and DNA.   

Cryptography: Congruence transformations are used in cryptography to encrypt and decrypt messages. They are used to perform operations such as permutation and substitution, which are essential for creating secure cryptographic systems.  

Congruence transformations are powerful mathematical tools with many applications in science, engineering, and other fields. For example, they are essential for understanding and modeling complex systems and solving problems requiring geometric and spatial reasoning.  

[juniperpublishers-ttsr]

Novel 3D visualization of Laplace and Fourier transforms

Abstract

This paper will demonstrate how to illustrate poles and zeros of a filter's transfer function as a 3D surface and how this facilitates an understanding of the relationship between the filter's amplitude response and the poles'/zeros' locations in s-space. The transfer function, in Laplace representation, is plotted as a function of the real and imaginary parts of the complex frequency s. This is exemplified for a Chebyshev-II low pass filter.

Keywords:    Laplace transform; Poles; zeros; Transfer function; Bode plot; MATLAB; Filter

Abbreviations: PBL: Problem-Based Learning; LTI: Linear Time-Invariant

    Introduction

Transforms are probably one of the most versatile engineering tools as they facilitate a visualization of signals in frequency space. The complexity of the subject is widely recognized by students and teachers in engineering disciplines all over the world. It has been suggested that this is due to inconsistencies in the portrayal of transforms in standard textbooks [1]. In order to further the understanding of this abstract subject, different approaches have been suggested. In [2], it was suggested that signals be treated as vectors and transforms should be introduced as a scalar product used to decompose signals in order to disclose signal properties veiled in time space. Other approaches includes PBL (Problem Based Learning) [3] and "conceptual labs” [4,5] and also the use of the proper mathematical tools has been investigated [6].

In this mini review on transforms, a seminal way to graphically visualize poles and zeros that occur in Laplace (and z) transforms as 3D surfaces is suggested. This also facilitates a very elegant means to demonstrate graphically the relationship between the Laplace transform and the Fourier transform.

    Discussion

In engineering, Laplace transforms appear for example, as the transfer function in a linear, time-invariant (LTI) system:

 where N(s) and D(s) are polynomials in the complex frequency variable s = s + jw [7]. The roots of the numerator are called zeros and the roots of the denominator are called poles. These are singularities in s-space; at every zero the transfer function has a zero value and at the poles the transfer function has an infinite value. If the polynomial is of order n, there are exactly n poles or zeros, respectively.

Since the frequency variable s is complex, it is a function of both the real part and the imaginary part ω ; H(σ) = H(σ,ω) and |H(σ,w)| is a positive surface in s-space where the poles and zeros will appear as singularities in the surface. For example, let's consider a 3rd order Chebyshev2, low-pass filter with stop band ripple equal to 3 dB:

 Figure 1 illustrates the Bode diagram of this filter.

The roots of the numerator and denominator are Zeros: 0 ± 1.1547j Poles: -0.1774 ± 1.0776j -3.3619 The poles and zeros are illustrated in a 2D plot in (Figure 2).

This is the traditional way of illustrating poles and zeros. A seminal approach is suggested as follows:

Rewrite the transfer function in (2) as a function of two variables; the real part and the imaginary part;

 Next, we need to find the absolute value H (σ, w). To this end, an online symbolic computational engine was used [8]. The resulting expression is a square root fraction of polynomials (that is not reproduced here due to limited space). This expression is then used to create a mesh grid in MATLAB which is used to plot a 3D color map of the H(σ, w) function. This is illustrated in (Figure 3) and in (Figure 4) we have zoomed in part of the surface.

First of all we can easily identify the three poles and the two zeros in Figure 3 (compare with Figure 2). Notice what the poles and zeros look like in 3D; poles pull the surface out toward infinity and zeros nail the surface to the “floor”.

The important thing here is to understand how all the graphs in (Figures 1,2 & 3) relate to each other. The relationship between (Figures 2 & 3) should be clear by just comparing the locations of the poles and zeros in the two figures. Relating Figure 1 to the other plots is harder.

The magnitude plot in (Figure 1) represents the filter's amplitude response, i.e. how it responds to harmonics of constant amplitude. In terms of the complex frequency variable s, this corresponds to frequencies with an imaginary part only. The real part of the complex frequency s = σ + jw represents harmonics with exponentially growing/decreasing amplitudes. Hence, the magnitude plot in (Figure 1) corresponds to the cross-section of the 3D surface σ = 0 . Also, in a Bode diagram, only positive frequencies are plotted. So, if we cut the 3D surface in Figure 3 at σ = 0 (= real axis; already done in Figure 3) ω = 0 and at (the imaginary axis), the cross-section edge of the remaining 3D surface is exactly the amplitude response of the filter. In Figure 4 we have plotted the remaining surface and zoomed in on the cross-section edge. Compare the surface edge and the magnitude plot in (Figure 1); they agree exactly.

    Conclusion

This work has suggested a seminal way to illustrate poles and zeros of a filter as a 3D plot with the objective of alleviating transform interpretation. From (Figures 3 & 4) the relationship between the pole/zero locations and the filter's amplitude response can be visualized.

It should also be pointed out that since the cross-section edge in Figures 3 & 4 represents σ= 0 , it also represents the magnitude of the Fourier transform, i.e. the Fourier transform of the filter's impulse response; the 3D representation of poles and zeros is also an excellent tool to demonstrate graphically the relationship between the Laplace transform and the Fourier transform.

The same exact technique could be applied to the z-transform with the only difference that the 3D surface should be cut along the unit circle to get the Fourier transform.

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[mamtajuyal1986]

IIT JAM 2020 Physics (PH) Syllabus | IIT JAM 2020 Physics (PH) Exam Pattern

IIT JAM 2020 Physics (PH) Syllabus | IIT JAM 2020 Physics (PH) Exam Pattern Physics (PH): The syllabus is a very important aspect while preparing for the examination. Therefore it is advised to all the appearing candidates that they should go through the Physics (PH) syllabus properly before preparing for the examination. Mathematical Methods: Calculus of single and multiple variables, partial derivatives, Jacobian, imperfect and perfect differentials, Taylor expansion, Fourier series. Vector algebra, Vector Calculus, Multiple integrals, Divergence theorem, Green's theorem, Stokes' theorem. First order equations and linear second order differential equations with constant coefficients. Matrices and determinants, Algebra of complex numbers. Mechanics and General Properties of Matter: Newton's laws of motion and applications, Velocity and acceleration in Cartesian, polar and cylindrical coordinate systems, uniformly rotating frame, centrifugal and Coriolis forces, Motion under a central force, Kepler's laws, Gravitational Law and field, Conservative and non-conservative forces. System of particles, Center of mass, equation of motion of the CM, conservation of linear and angular momentum, conservation of energy, variable mass systems. Elastic and inelastic collisions. Rigid body motion, fixed axis rotations, rotation and translation, moments of Inertia and products of Inertia, parallel and perpendicular axes theorem. Principal moments and axes. Kinematics of moving fluids, equation of continuity, Euler's equation, Bernoulli's theorem. Oscillations, Waves and Optics: Differential equation for simple harmonic oscillator and its general solution. Superposition of two or more simple harmonic oscillators. Lissajous figures. Damped and forced oscillators, resonance. Wave equation, traveling and standing waves in one-dimension. Energy density and energy transmission in waves. Group velocity and phase velocity. Sound waves in media. Doppler Effect. Fermat's Principle. General theory of image formation. Thick lens, thin lens and lens combinations. Interference of light, optical path retardation. Fraunhofer diffraction. Rayleigh criterion and resolving power. Diffraction gratings. Polarization: linear, circular and elliptic polarization. Double refraction and optical rotation. Electricity and Magnetism: Coulomb's law, Gauss's law. Electric field and potential. Electrostatic boundary conditions, Solution of Laplace's equation for simple cases. Conductors, capacitors, dielectrics, dielectric polarization, volume and surface charges, electrostatic energy. Biot-Savart law, Ampere's law, Faraday's law of electromagnetic induction, Self and mutual inductance. Alternating currents. Simple DC and AC circuits with R, L and C components. Displacement current, Maxwelll's equations and plane electromagnetic waves, Poynting's theorem, reflection and refraction at a dielectric interface, transmission and reflection coefficients (normal incidence only). Lorentz Force and motion of charged particles in electric and magnetic fields. Kinetic theory, Thermodynamics: Elements of Kinetic theory of gases. Velocity distribution and Equipartition of energy. Specific heat of Mono-, di- and tri-atomic gases. Ideal gas, van-der-Waals gas and equation of state. Mean free path. Laws of thermodynamics. Zeroth law and concept of thermal equilibrium. First law and its consequences. Isothermal and adiabatic processes. Reversible, irreversible and quasi-static processes. Second law and entropy. Carnot cycle. Maxwell's thermodynamic relations and simple applications. Thermodynamic potentials and their applications. Phase transitions and Clausius-Clapeyron equation. Ideas of ensembles, Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein distributions. Modern Physics: Inertial frames and Galilean invariance. Postulates of special relativity. Lorentz transformations. Length contraction, time dilation. Relativistic velocity addition theorem, mass energy equivalence. Blackbody radiation, photoelectric effect, Compton effect, Bohr's atomic model, X-rays. Wave-particle duality, Uncertainty principle, the superposition principle, calculation of expectation values, Schrodinger equation and its solution for one, two and three dimensional boxes. Solution of Schrodinger equation for the one dimensional harmonic oscillator. Reflection and transmission at a step potential, Pauli exclusion principle. Structure of atomic nucleus, mass and binding energy. Radioactivity and its applications. Laws of radioactive decay. Solid State Physics, Devices and Electronics: Crystal structure, Bravais lattices and basis. Miller indices. X-ray diffraction and Bragg's law; Intrinsic and extrinsic semiconductors, variation of resistivity with temperature. Fermi level. p-n junction diode, I-V characteristics, Zener diode and its applications, BJT: characteristics in CB, CE, CC modes. Single stage amplifier, two stage R-C coupled amplifiers. Simple Oscillators: Barkhausen condition, sinusoidal oscillators. OPAMP and applications: Inverting and non-inverting amplifier. Boolean algebra: Binary number systems; conversion from one system to another system; binary addition and subtraction. Logic Gates AND, OR, NOT, NAND, NOR exclusive OR; Truth tables; combination of gates; de Morgan's theorem. Related Articles: IIT JAM 2020 Syllabus Biotechnology (BT) Syllabus Biological Sciences (BL) Syllabus Chemistry (CY) Syllabus Geology (GG) Syllabus Mathematics (MA) Syllabus Mathematical Statistics (MS) Syllabus Physics (PH) Syllabus Read the full article

[coursesforallacademynoida]

B.Sc Tuition In Noida For Ordinary Differential equations

B.Sc Tuition In Noida For Ordinary Differential equations

B.Sc Tuition In Noida For Ordinary Differential equations

CFA Academy For B.Sc Maths Tuition Classes In Noida. Linear Algebra Tuition In Noida, Calculus Tuition In Noida, Real Analysis Tuition In Noida, Complex Analysis Tuition In Noida, Ordinary Differential Equations Tuition In Noida, Algebra Tuition In Noida, Functional Analysis Tuition In Noida, Numerical Analysis Tuition In Noida, Partial…

[arxt1]

The use of a Kepler solver in numerical integrations of quasi-Keplerian orbits. (arXiv:1911.02773v2 [gr-qc] UPDATED)

A Kepler solver is an analytical method to solve a pure two-body problem. By slightly modifying the Kepler solver, we develop a new manifold correction method. Only one change to the analytical solution lies in that the obtainment of the eccentric anomaly does not need any iteration but uses the true anomaly between the constant Laplace-Runge-Lenz (LRL) vector and a varying radial vector given by an integrator. Such a new method rigorously conserves seven integrals of the Kepler energy, angular momentum vector and LRL vector. This leads to the numerical conservation of all orbital elements except the mean longitude. In the construction mechanism, the new method is unlike the existing Fukushima's linear transformation method that satisfies these properties. It can be extend to treat quasi-Keplerian orbits in perturbed two-body or N-body problems. The five slowly-varying orbital elements of each body relative to the central body are determined by the seven quasi-conserved quantities from their integral invariant relations, and the eccentric anomaly is calculated similarly in the above way. Substituting these values into the Kepler solver yields an adjusted solution of each body. Taking a post-Newtonian two-body problem and a six-body system of the Sun, four outer planets and Pluto as examples of quasi-Keplerian motions, we show that the new method can significantly improve the accuracies of all the orbital elements and the positions of individual planets, as compared with the case without correction. The new method and the Fukushima's method are almost the same in the numerical performance and need small additional computational cost.

from gr-qc updates on arXiv.org https://ift.tt/2CyIcG6

[digitalcontentmarketorg-blog]

Download Full Solution Manual for Differential Equations an Introduction to Modern Methods and Applications 3rd Edition By

Solution Manual for Differential Equations An Introduction to Modern Methods and Applications 3rd Edition by Brannan and Boyce

Link download full: https://getbooksolutions.com/download/solution-manual-for-differential-equations-an-introduction-to-modern-methods-and-applications-3rd-edition-by-brannan/

CLICK HERE TO VIEW SAMPLE OF Differential Equations An Introduction to Modern Methods and Applications 3rd Edition Solution manual by James R. Brannan and Boyce

Brannan/Boyce’s Differential Equations: An Introduction to Modern Methods and Applications, 3rd Edition is consistent with the way engineers and scientists use mathematics in their daily work. The text emphasizes a systems approach to the subject and integrates the use of modern computing technology in the context of contemporary applications from engineering and science. The focus on fundamental skills, careful application of technology, and practice in modeling complex systems prepares students for the realities of the new millennium, providing the building blocks to be successful problem-solvers in today’s workplace. Section exercises throughout the text provide hands-on experience in modeling, analysis, and computer experimentation. Projects at the end of each chapter provide additional opportunities for students to explore the role played by differential equations in the sciences and engineering.

Table of Contents

Chapter 1: Introduction

1.1 Mathematical Models and Solutions

1.2 Qualitative Methods: Phase Lines and Direction Fields

1.3 Definitions, Classification, and Terminology

Chapter 2: First Order Differential Equations

2.1 Separable Equations

2.2 Linear Equations: Method of Integrating Factors

2.3 Modeling with First Order Equations

2.4 Differences Between Linear and Nonlinear Equations

2.5 Autonomous Equations and Population Dynamics

2.6 Exact Equations and Integrating Factors

2.7 Substitution Methods

Projects

2.P.1 Harvesting a Renewable Resource

2.P.2 A Mathematical Model of a Groundwater Contaminant Source

2.P.3 Monte Carlo Option Pricing: Pricing Financial Options by Flipping a Coin

Chapter 3: Systems of Two First Order Equations

3.1 Systems of Two Linear Algebraic Equations

3.2 Systems of Two First Order Linear Differential Equations

3.3 Homogeneous Linear Systems with Constant Coefficients

3.4 Complex Eigenvalues

3.5 Repeated Eigenvalues

3.6 A Brief Introduction to Nonlinear Systems

Projects

3.P.1 Estimating Rate Constants for an Open Two-Compartment Model

3.P.2 A Blood-Brain Pharmacokinetic Model

Chapter 4: Second Order Linear Equations

4.1 Definitions and Examples

4.2 Theory of Second Order Linear Homogeneous Equations

4.3 Linear Homogeneous Equations with Constant Coefficients

4.4 Mechanical and Electrical Vibrations

4.5 Nonhomogeneous Equations; Method of Undetermined Coefficients

4.6 Forced Vibrations, Frequency Response, and Resonance

4.7 Variation of Parameters

Projects

4.P.1 A Vibration Insulation Problem

4.P.2 Linearization of a Nonlinear Mechanical System

4.P.3 A Spring-Mass Event Problem

4.P.4 Euler-Lagrange Equations

Chapter 5: The Laplace Transform

5.1 Definition of the Laplace Transform

5.2 Properties of the Laplace Transform

5.3 The Inverse Laplace Transform

5.4 Solving Differential Equations with Laplace Transforms

5.5 Discontinuous Functions and Periodic Functions

5.6 Differential Equations with Discontinuous Forcing Functions

5.7 Impulse Functions

5.8 Convolution Integrals and Their Applications

5.9 Linear Systems and Feedback Control

Projects

5.P.1 An Electric Circuit Problem

5.P.2 The Watt Governor, Feedback Control, and Stability

Chapter 6: Systems of First Order Linear Equations

6.1 Definitions and Examples

6.2 Basic Theory of First Order Linear Systems

6.3 Homogeneous Linear Systems with Constant Coefficients

6.4 Nondefective Matrices with Complex Eigenvalues

6.5 Fundamental Matrices and the Exponential of a Matrix

6.6 Nonhomogeneous Linear Systems

6.7 Defective Matrices

Projects

6.P.1 Earthquakes and Tall Buildings

6.P.2 Controlling a Spring-Mass System to Equilibrium

Chapter 7: Nonlinear Differential Equations and Stability

7.1 Autonomous Systems and Stability

7.2 Almost Linear Systems

7.3 Competing Species

7.4 Predator-Prey Equations

7.5 Periodic Solutions and Limit Cycles

7.6 Chaos and Strange Attractors: The Lorenz Equations

Projects

7.P.1 Modeling of Epidemics

7.P.2 Harvesting in a Competitive Environment

7.P.3 The Rossler System

Chapter 8: Numerical Methods

8.1 Numerical Approximations: Euler’s Method

8.2 Accuracy of Numerical Methods

8.3 Improved Euler and Runge-Kutta Methods

8.4 Numerical Methods for Systems of First Order Equations

Projects

8.P.1 Designing a Drip Dispenser for a Hydrology Experiment

8.P.2 Monte Carlo Option Pricing: Pricing Financial Option by Flipping a Coin

Chapter 9: Series Solutions of Second order Equations

9.1 Review of Power Series

9.2 Series Solutions Near an Ordinary Point, Part I

9.3 Series Solutions Near an Ordinary Point, Part II

9.4 Regular Singular Points

9.5 Series Solutions Near a Regular Singular Point, Part I

9.6 Series Solutions Near a Regular Singular Point, Part II

9.7 Bessel’s Equation

Projects

9.P.1 Diffraction Through a Circular Aperature

9.P.2 Hermite Polynomials and the Quantum Mechanical Harmonic Oscillator

9.P.3 Perturbation Methods

Chapter 10: Orthogonal Functions, Fourier Series and Boundary-Value Problems

10.1 Orthogonal Families in the Space PC [a,b]

10.2 Fourier Series

10.3 Elementary Two-Point Boundary Value Problems

10.4 General Sturm-Liouville Boundary Value Problems

10.5 Generalized Fourier Series and Eigenfunction Expansions

10.6 Singular Boundary Value Problems

10.7 Convergence Issues

Chapter 11: Elementary Partial Differential Equations

11.1 Terminology

11.2 Heat Conduction in a Rod—Homogeneous Case

11.3 Heat Conduction in a Rod—Nonhomogeneous Case

11.4 Wave Equation—Vibrations of an Elastic String

11.5 Wave Equation—Vibrations of a Circular Membrane

11.6 Laplace Equation

Projects

11.P.1 Estimating the Diffusion Coefficient in the Heat Equation

11.P.2 The Transmission Line Problem

11.P.3 Solving Poisson’s Equation by Finite Differences

11.P.4 Dynamic Behavior of a Hanging Cable

11.P.5 Advection Dispersion: A Model for Solute Transport in Saturated Porous Media

11.P.6 Fisher’s Equation for Population Growth and Dispersion

Appendices

11.A Derivation of the Heat Equation

11.B Derivation of the Wave Equation

A: Matrices and Linear Algebra

A.1 Matrices

A.2 Systems of Linear Algebraic Equations, Linear Independence, and Rank

A.3 Determinates and Inverses

A.4 The Eigenvalue Problem

B: Complex Variables

ISBN-13:  978-1118531778 ISBN-10:  1118531779

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[physical-laws-en-blog]

Superposition principle

This article is about the superposition principle in linear systems. For other uses, see Superposition (disambiguation). Superposition of almost plane waves (diagonal lines) from a distant source and waves from the wake of the ducks. Linearity holds only approximately in water and only for waves with small amplitudes relative to their wavelengths. In physics and systems theory, the superposition principle, also known as superposition property, states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually. So that if input A produces response X and input B produces response Y then input (A + B) produces response (X + Y). The homogeneity and additivity properties together are called the superposition principle. A linear function is one that satisfies the properties of superposition. It is defined as F ( x 1 + x 2 ) = F ( x 1 ) + F ( x 2 ) {\displaystyle F(x_{1}+x_{2})=F(x_{1})+F(x_{2})\,}  Additivity F ( a x ) = a F ( x ) {\displaystyle F(ax)=aF(x)\,}  Homogeneity for scalar a. This principle has many applications in physics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, frequency domain linear transform methods such as Fourier, Laplace transforms, and linear operator theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behaviour. The superposition principle applies to any linear system, including algebraic equations, linear differential equations, and systems of equations of those forms. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, or any other object which satisfies certain axioms. Note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum. More details Android, Windows

[dnsiaody]

CSIR UGC NET Application 2017

COMMON SYLLABUS FOR PART ‘B’ PLUS ‘C’ MATHEMATICAL SCIENCES DEVICE - 1 Analysis: Primary set theory, finite, countable and uncountable sets, True number system as the complete ordered field, Archimedean property, supremum, infimum. admission.scholarshipbag.com Sequences and series, convergence, limsup, liminf. Bolzano Weierstrass theorem, Heine Borel theorem. Continuity, uniform continuity, differentiability, indicate value theorem. Sequences plus a number of functions, homogeneous convergence. Riemann sums plus Riemann integral, Improper Integrals. Monotonic functions, types associated with discontinuity, functions of bounded variation, Lebesgue measure, Lebesgue integral. Functions of various variables, directional derivative, partially derivative, derivative as being a geradlinig transformation, inverse and implied function theorems. Metric areas, compactness, connectedness. Normed geradlinig Spaces. Spaces of constant functions as examples. Geradlinig Algebra: Vector spaces, subspaces, linear dependence, basis, aspect, algebra of linear changes. Algebra of matrices, position and determinant of matrices, linear equations. Eigenvalues plus eigenvectors, Cayley-Hamilton theorem. Matrix representation of linear changes. Change of basis, canonical forms, diagonal forms, triangular forms, Jordan forms. Internal product spaces, orthonormal base. Quadratic forms, reduction plus classification of quadratic types UNIT - two Complicated Analysis: Algebra of complicated numbers, the complex aircraft, polynomials, power series, transcendental functions such as rapid, trigonometric and hyperbolic features. Analytic functions, Cauchy-Riemann equations. Contour integral, Cauchy’s theorem, Cauchy’s integral formula, Liouville’s theorem, Maximum modulus rule, Schwarz lemma, Open umschlüsselung theorem. Taylor series, Laurent series, calculus of residues. Conformal mappings, Mobius changes. Algebra: Permutations, combinations, pigeon-hole principle, inclusion-exclusion principle, derangements. Fundamental theorem of math, divisibility in Z, congruences, Chinese Remainder Theorem, Euler’s Ø- function, primitive origins. Groups, subgroups, normal subgroups, quotient groups, homomorphisms, cyclic groups, permutation groups, Cayley’s theorem, class equations, Sylow theorems. Rings, ideals, excellent and maximal ideals, quotient rings, unique factorization site, principal ideal domain, Euclidean domain. Polynomial rings plus irreducibility criteria. Fields, limited fields, field extensions, Galois Theory. Topology: basis, thick sets, subspace and item topology, separation axioms, connectedness and compactness. UNIT -- 3 Ordinary Differential Equations (ODEs): Existence and originality of solutions of preliminary value problems for initial order ordinary differential equations, singular solutions of initial order ODEs, a system associated with first order ODEs. The common theory of homogenous plus nonhomogeneous linear ODEs, deviation of parameters, Sturm-Liouville border value problem, Green’s functionality. Partial Differential Equations (PDEs): Lagrange and Charpit strategies for solving first purchase PDEs, Cauchy problem intended for first order PDEs. Category of second order PDEs, General solution of increased order PDEs with continuous coefficients, Method of splitting up of variables for Laplace, Heat and Wave equations. Numerical Analysis: Numerical options of algebraic equations, Technique of iteration and Newton-Raphson method, Rate of convergence, Solution of systems associated with linear algebraic equations making use of Gauss elimination and Gauss-Seidel methods, Finite differences, Lagrange, Hermite and spline interpolation, Numerical differentiation and incorporation, Numerical solutions of ODEs using Picard, Euler, customised Euler andRunge-Kutta methods. Calculus of Variations: A variety associated with a functional, Euler-Lagrange formula, Necessary and sufficient situations for extreme. Variational strategies for boundary value troubles in ordinary and partially differential equations. Linear Essential Equations: The Linear integral formula of the first plus second kind of Fredholm and Volterra type, Options with separable kernels. Feature numbers and eigenfunctions, resolvent kernel. Classical Mechanics: General coordinates, Lagrange’s equations, Hamilton’s canonical equations, Hamilton’s rule and the principle of minimum action, Two-dimensional motion associated with rigid bodies, Euler’s dynamical equations for the movement of the rigid entire body about an axis, the concept of small oscillations. DEVICE - four Descriptive figures, exploratory data analysis Example space, discrete probability, 3rd party events, Bayes theorem. Unique variables and distribution features (univariate and multivariate); requirement and moments. Independent unique variables, marginal and conditional distributions. Characteristic functions. Possibility inequalities (Tchebyshef, Markov, Jensen). Modes of convergence weakened and strong laws associated with large numbers, Central Restrict theorems (i. i. g. case). Markov chains along with finite and countable condition space, classification of claims, limiting behaviour of n-step transition probabilities, stationary submission, Poisson and birth-and-death procedures. Standard discrete and constant univariate distributions. sampling distributions, standard errors and asymptotic distributions, distribution of purchase statistics and range. Strategies of estimation, properties associated with estimators, confidence intervals. Testing of hypotheses: most effective and uniformly most effective tests, likelihood ratio testing. Analysis of discrete information and chi-square test associated with goodness of fit. Big sample tests. Simple non-parametric tests for just one particular and two sample troubles, rank correlation and check for independence. Elementary Bayesian inference. Gauss-Markov models, estimability of parameters, best geradlinig unbiased estimators, confidence times, tests for linear ideas. Analysis of variance plus covariance. Fixed, random plus mixed effects models. Assured multiple linear regression. Primary regression diagnostics. Logistic regression. Multivariate normal distribution, Wishart distribution and their qualities. Distribution of quadratic types. Inference for parameters, partially and multiple correlation coefficients and related tests. Information reduction techniques: Principle element analysis, Discriminant analysis, Bunch analysis, Canonical correlation. Easy random sampling, stratified sample and systematic sampling. Possibility proportional to size sample. Ratio and regression strategies. Completely randomised designs, randomised block designs and Latin-square designs. Connectedness and orthogonality of block designs, BIBD. 2K factorial experiments: confounding and construction. Hazard functionality and failure rates, censoring and life testing, collection and parallel systems. Geradlinig programming problem, simple strategies, duality. Elementary queuing plus inventory models. Steady-state options of Markovian queuing versions: M/M/1, M/M/1 with restricted waiting space, M/M/C, M/M/C with the limited waiting area, M/G/1. All students are usually required to answer queries from Unit I. College students in mathematics are anticipated to answer an additional query from Unit II plus III. Students with within statistics are required in order to answer the additional question through Unit IV.

[arxt1]

Use of a Kepler solver in N-body simulations. (arXiv:1911.02773v1 [gr-qc])

A Kepler solver is an analytical method to solve a two-body problem. It is slightly modified as a new manifold correction method. Only one change to the analytical solution lies in that the obtainment of the eccentric anomaly does not need any iteration but uses the true anomaly between the constant Laplace-Runge-Lenz (LRL) vector and a varying radial vector given by an integrator. The new method rigorously conserves seven integrals of the Kepler energy, angular momentum vector and LRL vector. This leads to the conservation of all orbital elements except the mean longitude. In the construction mechanism, the new method is unlike the existing Fukushima's linear transformation method that satisfies these properties. It can be extend to treat an $N$-body problem. The five slowly-varying orbital elements of each body relative to the central body are determined by the seven quasi-conserved quantities from their integral invariant relations, and the eccentric anomaly is calculated similarly in the above way. Substituting these values into the Kepler solver yields an adjusted solution of each body. Numerical simulations of a six-body problem of the Sun, Jupiter, Saturn, Uranus, Neptune and Pluto show that the new method can significantly improve the accuracy of all the orbital elements and positions of individual planets, as compared with the case without correction. The new method and the Fukushima's method are almost the same in the numerical performance and require negligibly additional computational cost. They both are the best of the existing correction methods in $N$-body simulations.

from gr-qc updates on arXiv.org https://ift.tt/2CyIcG6