Journal paper available here: TauFactor is a **MatLab** application for efficiently calculating the tortuosity factor, as well as **volume** fractions, surface areas and triple phase boundary densities, from image based microstructural data. The tortuosity factor quantifies the apparent decrease in diffusive.

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For example, if the plaintext message is 12345678ABCDEF12 and the key to encrypt the plaintext be 1234123412341234, then the cipher text will be E112BE1DEFC7A367. Decryption of the above cipher text using the same key results in the plain text message 12345678ABCDEF12 .

### Unlike the finite-element method, no fundamental solution is needed

Round key used in encryption process. Each block of 64-bit of ciphered image is the input for the DES. After 16 rounds of processing, the final output will be 64-bit plaintext. Similarly same process is repeated for all the remaining ciphered blocks. The original image is recovered after storing all the decrypted plaintexts of size x*y.

In mathematics, finite-difference methods (FDM) are numerical methods for solving differential equations by approximating them with difference equations, in which finite differences approximate the derivatives. FDMs are thus discretization methods (helpful site). FDMs convert a linear (non-linear) ODE/PDE into a system of linear (non-linear) equations, which can then be solved by matrix algebra techniques. The reduction of the differential equation to a system of algebraic equations makes the problem of finding the solution to a given ODE ideally suited to modern computers, hence the widespread use of FDMs in modern numerical analysis.

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Where, a and b are in rational numbers, complex numbers, integers mod n. In the above equation, if 43 + 27 2 0 ,the equation represents a nonsingular elliptic curve; otherwise, the equation represented a singular elliptic curve. In a nonsingular elliptic curve, the equation 3 + + = 0 has three distinct roots (real or complex); in a singular curve the equation 3 + + = 0 does not have three distinct roots .

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Here, we have chosen p (modulo) = 257 for ECC. The value of a and b of elliptical curve are 6 & 7 respectively.

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The modern world is experiencing multimedia and internet usage techniques. As, image data distribution is increasing, the security risks and threats are also coming into picture. To avoid the risks and to make the transmission of images secure, digital encryption techniques are evolved over recent years. In this paper, a digital image encryption technique has been proposed and simulated to enhance the security performance, efficiency and reliability. Original image is encrypted using DES (Digital Encryption Standard) with the help of a key sequence which is generated from a chaotic key generator with the help of Henon map and the encrypted image is mapped to the points of elliptic curve.

Fosite is a generic framework for the numerical solution of hyperbolic conservation laws in generalized orthogonal coordinates. Its main purpose is the simulation of compressible flows in accretion disks. The underlying numerical solution method belongs to the family of unsplit conservative finite volume TVD schemes. The method (navigate here) is 2nd order accurate in space and uses high order Runge-Kutta and multistep schemes for time evolution. In addition to the pure advection code several source terms have.

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**Methods** of discretizing the equations are discussed and transformation techniques and grids are presented. Two examples of numerical methods close out this part of the book: source and vortex panel methods and the explicit method. Part II is devoted to four self-contained chapters on more advanced material. Roger Grundmann treats the boundary layer equations and methods of solution.

### Numerical and Statistical Methods

The Henon map was proposed by M. Henon in 1976 which is 2D map represented with an attractor and also is a Lorenz equation of Poincare section. The Henon map generates cipher key which is random sequence and is used to encrypt the shuffled image or plain image. Chaotic behavior depends on initial system parameters and conditions.

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I will also appreciate it if you write me a couple of lines about how you have used it in your research. It encourages me to maintain the code.

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MATLAB code that generates all figures in [Almquist and Dunham, 2021, ]. The paper considers narrow-stencil summation-by-parts finite difference methods and derives new penalty terms for boundary and interface conditions. The new penalty terms are significantly less stiff than the previous state-of-the-art method on curvilinear grids. On highly skewed grids, there may be an order of magnitude or more difference in the time-step restriction.

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The Data Encryption Standard algorithm (DES) is the most widely used symmetric encryption algorithm in the world so far. DES was quickly adopted for non- digital media. The banking industry, adopted DES as a banking standard. Standards for the banking industry are set by the American National Standards Institute . DES is a symmetric block cipher designed to encrypt and decrypt blocks of data consisting of 64 bits under control of a 56 bit key. Each 8th bit of the 64-bit key is used for parity checking and otherwise ignored. Decrypting must be done by using the same key as for encryption.

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In past years, the algorithms of many cryptosystems are under development and enhancement. DES is still important, so improvements and accuracy is required. In this paper, we have taken a grayscale image and encrypted it with DES. The round key is generated from Henon chaotic map. The resulting image is further mapped using elliptic curve, which enhances the complexity of algorithm and impossible to attack without knowing the system parameters. In future, we will work with variations of DES along with 3D chaotic map.

The author promotes an understanding of common terminology with a balance between mathematical rigor and physical intuition that characterizes the origin of the methods. This book aims to be a first contact with finite volume methods.

### Numerical Methods for Partial Differential Equations

Liu et. al. had proposed a new method to enhance the algorithm of DES. In DES, logistic map is used to generate the random round keys which further enhance the complexity of the algorithm. The experiment in the paper shows that after image encryption, the original image becomes stochastic noise which is impossible for the attacker to attack.

In each round a function, f is performed in which the data is combined with a 48-bit permutation of the key. After the 16th iteration, the right (R16) and left (L16) halves are concatenated and a inverse permutation (IP-1) completes the encryption process . The figure for the entire DES encryption is shown Figure 1 .

In present, information security is very important in digital communication. As the internet and multimedia technology increasing, a need of secure algorithm is required to protect the authenticated and authorized multimedia contents such as, image, audio and video etc. Sensitive information like medical and legal records, credit ratings, Business transactions, Voice mail are also require to be protected from outsiders.

So, clamp the pressure of one node in the domain to zero as a boundary condition and the pressure at other nodes will be measured relative to that point. It's a common practice to set P(1,1)=0 and because P(1,1) is known P'(1,1)=0, but you'll need to compute P' at other nodes. It's obvious that aS = 0 for a P' point that is located on the bottom wall because there is no P' node under it but it has aW,aE,aN.

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Numerical methods, as said above are techniques to approximate mathematical procedures. On the other hand, statistics is the study and manipulation of data, including ways to gather, review, analyze, and draw conclusions from the given data. Thus we can say, statistical methods are mathematical formulas, models, and techniques that are used in the statistical analysis of raw research data. The application of statistical methods extracts information from research data and provides different methods to assess the robustness of research outputs.

The result passes into 8 S-Boxes. Each S-Box transforms 6 bit input to 4 bit output.

### Types of Numerical Methods

The initial value of A0, B0 is replaced by A1, B1 in next iteration of generation of sequence in Henon map. The Henon attractor is used to converge the sequence. The u and v are control parameters. For u=1/4 and v=0/3, the Henon map generates random sequence with chaotic behavior .

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Hardware Simulation and Vitualization Environment: Unfortunately I have lost my source code in a freak accident (My roommate dropped a bed onto my laptop) for this project. Please be patient and visit for more.

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### Projected finite elements for reaction–diffusion systems on stationary closed surfaces

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