How Light Enters the Eye

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Refraction of Light

Refraction is the change in direction of a wave due to a change in its speed. This is most commonly seen when a wave passes from one medium to another. Refraction of light is the most commonly seen example, but any type of wave can refract when it interacts with a medium, for example when sound waves pass from one medium into another or when water waves move into water of a different depth. In optics,...

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Laws of reflection

Reflection is the change in direction of a wave front at an interface between two different media so that the wave front returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves.Reflection of light may be specular (that is, mirror-like) or diffuse (that is, not retaining the image, only the energy) depending on the nature of the interface....

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Prism

Prism is a transparent optical element with flat, polished surfaces that refract light. The exact angles between the surfaces depend on the application. The traditional geometrical shape is that of a triangular prism with a triangular base and rectangular sides, and in colloquial use "prism" usually refers to this type. Some types of optical prism are not in fact in the shape of geometric prisms....

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SPECTRUM

A spectrum (plural spectra or spectrums) is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by analogy to many fields. Thus one might talk about the spectrum of political opinion,...

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Modern Physics: Special Relativity

Special relativity (SR) (also known as the special theory of relativity or STR) is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein (after considerable contributions of Hendrik Lorentz and Henri Poincaré) in the paper "On the Electrodynamics of Moving Bodies". It generalizes Galileo's principle of relativity – that all uniform motion is relative,...

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Resolution - Refutation Proofs

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Programming Paradigms

Computer architecture by discussing function call and return in further depth as well as introducing the swap function in relation to C++ c...

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computer architecture function call and returns in relation to C code.

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Computer architecture and the Assembly programming language

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Heap segments and their use in the C programming language

HEap Swgment is a segment of memory claimed by a program. Within the heap, pieces of memory can be used and freed as needed. Depending on usage, heaps can become fragmented as small pieces of memory are used and freed. In formal computer science terms, a heap is the same as a partially ordered tr...

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C programming language and generic stacks

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Programming Paradigms-C language programming by focusing on different forms of stack.

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Linear Dynamical Systems Lecture 20-Observability and State Estimation

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Lecture 19 Controllability and State Transfer and their uses in modern electrical engineering

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Linear Dynamical Systems Lecture 18 -Applications of SVD, Controllability,and State Transfer in Electrical Engineering

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Linear Dynamical Systems Lecture 17-Applications of single value decomposition in LDS and electrical engineering

Singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. Applications which employ the SVD include computing the pseudoinverse, least squares fitting of data, matrix approximation, and determining the rank, range and null space of a matr...

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Linear Dynamical Systems Lecture 16- Use of symmetric matrices, quadratic forms, matrix norm, and SVDs in LDS

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Linear Dynamical Systems Lecture 15-Inputs and Outputs of symmetric matrices

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Linear Dynamical Systems Lecture 14- Applications of Jordan Canonical Form in LDS and Electrical Engineering

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Linear Dynamical Systems Lecture 13-generalized eigenvectors, diagonalization, and Jordan canonical form

Jordan normal form (often called Jordan canonical form) shows that a given square matrix M over a field K containing the eigenvalues of M can be transformed into a certain normal form by changing the basis. This normal form is almost diagonal in the sense that its only non-zero entries lie on the diagonal and the superdiagonal. This is made more precise in the Jordan-Chevalley decomposition. One can...

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Linear Dynamical Systems Lecture12-Matrix exponentials,Eigenvectors,and Diagonalization and their uses in LDS

Matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group.In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e. if there exists an invertible matrix P such that P −1AP is a diagonal...

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Linear Dynamical Systems Lecture 11-LaPlace transform use of matrix exponentials

Matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group.Laplace transform is one of the best known and widely used integral transforms. It is commonly used to produce an easily solvable algebraic equation from an ordinary differential...

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Linear Dynamical Systems Lecture 9&10 -Autonomous linear dynamical systems

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Linear Dynamical Systems Lecture 8-Least Norm solutions of undetermined equations

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Linear Dynamical Systems Lecture 7-Regularized least squares and the Gauss-Newton method

Gauss–Newton algorithm is a method used to solve non-linear least squares problems. It can be seen as a modification of Newton's method for finding a minimum of a function. Unlike Newton's method, the Gauss–Newton algorithm can only be used to minimize a sum of squared function values, but it has the advantage that second derivatives, which can be challenging to compute, are not required.Non-linear...

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Linear Dynamical Systems Lecture 6-Applications of least squares

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Linear Dynamical Systems Lecture 5- QR Factorization and least squares

The method of least squares is used to solve overdetermined systems. Least squares is often applied in statistical contexts, particularly regression analysis.Least squares can be interpreted as a method of fitting data. The best fit in the least-squares sense is that instance of the model for which the sum of squared residuals has its least value, a residual being the difference between an observed...

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Linear Dynamical Systems lecture 4-Orthonormal sets of vectors and QR factorization

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and both of unit length. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal bas...

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Linear Dynamical Systems Lecture 3-Linear algebra

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis. Linear algebra also has a concrete representation...

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Linear Dynamical Systems Lecture 2 -Linear functions

A linear function is a real or complex function f with the functional equation y=f(x)=m⋅x+b, where m and b are real (or complex) numbers. The equation y=f(x)=m⋅x+b is called (general) linear equation. The graph of a real linear function is a straight li...

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Introduction to Linear Dynamical Systems Lecture 1

Introduction to applied linear algebra and linear dynamical systems, with applications to circuits, signal processing, communications, and control systems. Topics include: Least-squares aproximations of over-determined equations and least-norm solutions of underdetermined equations. Symmetric matrices, matrix norm and singular value decomposition. Eigenvalues, left and right eigenvectors, and dynamical...

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Convex Optimization II

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Knowledge Based Systems: Logic and Deduction

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Convex Optimization

Convex optimization is a subfield of mathematical optimization. Given a real vector space together with a convex, real-valued functiondefined on a convex subset of , the problem is to find the point in for which the number is smallest, i.e., the point such that for all .The convexity of and make the powerful tools of convex analysis applicable: the Hahn–Banach theorem and the theory of subgradients...

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Programming Abstractions

The computing environment of tomorrow will not be anything like the stand-alone environment of yesterday. Computing infrastructure is changing drastically, presenting unique opportunities as well as challenges for applications and application scientists. Despite the rapid onset of distributed infrastructure, the majority of scientists still use the same programming methods that they are accustomed...

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Special Relativity Lecture

Special relativity (SR) (also known as the special theory of relativity or STR) is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein (after considerable contributions of Hendrik Lorentz and Henri Poincaré) in the paper "On the Electrodynamics of Moving Bodies". It generalizes Galileo's principle of relativity – that all uniform motion is relative,...

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