Fourier series represent periodic functions as the sum of sines and cosines. They break functions down into simple terms that can be solved individually and then recombined to approximate the original function to any desired accuracy. Fourier series are written as the sum of a constant term and coefficients multiplied by sine and cosine terms, with the coefficients of either sines or cosines being zero if the original function is even or odd, respectively.
The Mean Value Theorem states that for any given curve between two endpoints, there must be a point at which the slope of the tangent to the curve is same as the slope of the secant through its endpoints. Copy the link given below and paste it in new browser window to get more information on Mean Value Theorem www.askiitians.com/iit-jee-applications-of-derivatives/rolle-theoram-and-lagrange-mean-value-theorem/
The Mean Value Theorem states that for any given curve between two endpoints, there must be a point at which the slope of the tangent to the curve is same as the slope of the secant through its endpoints. Copy the link given below and paste it in new browser window to get more information on Mean Value Theorem www.askiitians.com/iit-jee-applications-of-derivatives/rolle-theoram-and-lagrange-mean-value-theorem/
A polynomial interpolation algorithm is developed using the Newton's divided-difference interpolating polynomials. The definition of monotony of a function is then used to define the least degree of the polynomial to make efficient and consistent the interpolation in the discrete given function. The relation between the order of monotony of a particular function and the degree of the interpolating polynomial is justified, analyzing the relation between the derivatives of such function and the truncation error expression. In this algorithm there is not matter about the number and the arrangement of the data points, neither if the points are regularly spaced or not. The algorithm thus defined can be used to make interpolations in functions of one and several dependent variables. The algoritm automatically select the data points nearest to the point where an interpolation is desired, following the criterion of symmetry. Indirectly, the algorithm also select the number of data points, which is a unity higher than the order of the used polynomial, following the criterion of monotony. Finally, the complete algoritm is presented and subroutines in fortran code is exposed as an addendum. Notice that there is not the degree of the interpolating polynomial within the arguments of such subroutines.
On Generalized Classical Fréchet Derivatives in the Real Banach SpaceBRNSS Publication Hub
In this work, we reviewed the Fréchet derivatives beginning with the basic definitions and touching most of the important basic results. These results include among others the chain rule, mean value theorem, and Taylor’s formula for differentiation. Obviously, having clarified that the Fréchet differential operators exist in the real Banach domain and that the operators are clearly continuous, we then in the last section for main results developed generalized results for the Fréchet derivatives of the chain rule, mean value theorem, and Taylor’s formula among others which become highly useful in the analysis of generalized Banach space problems and their solutions in Rn.
Sentient Arithmetic and Godel's Incompleteness TheoremsKannan Nambiar
For me, there is only one logic that we rational human beings are able to accept and appreciate, and that is the mathematical logic of ZF theory. But in the last century we found that ZF theory is not in a position to provide all that we want, and went in search of a new mode of thinking and got one which we called meta mathematics. My question is: if we can put the unambiguous logic of ZF theory on paper, why can't we do the same with meta mathematics. This paper is my feeble attempt in that direction.
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
MEET US AT:
www.anuragtyagiclasses.com
A polynomial interpolation algorithm is developed using the Newton's divided-difference interpolating polynomials. The definition of monotony of a function is then used to define the least degree of the polynomial to make efficient and consistent the interpolation in the discrete given function. The relation between the order of monotony of a particular function and the degree of the interpolating polynomial is justified, analyzing the relation between the derivatives of such function and the truncation error expression. In this algorithm there is not matter about the number and the arrangement of the data points, neither if the points are regularly spaced or not. The algorithm thus defined can be used to make interpolations in functions of one and several dependent variables. The algoritm automatically select the data points nearest to the point where an interpolation is desired, following the criterion of symmetry. Indirectly, the algorithm also select the number of data points, which is a unity higher than the order of the used polynomial, following the criterion of monotony. Finally, the complete algoritm is presented and subroutines in fortran code is exposed as an addendum. Notice that there is not the degree of the interpolating polynomial within the arguments of such subroutines.
On Generalized Classical Fréchet Derivatives in the Real Banach SpaceBRNSS Publication Hub
In this work, we reviewed the Fréchet derivatives beginning with the basic definitions and touching most of the important basic results. These results include among others the chain rule, mean value theorem, and Taylor’s formula for differentiation. Obviously, having clarified that the Fréchet differential operators exist in the real Banach domain and that the operators are clearly continuous, we then in the last section for main results developed generalized results for the Fréchet derivatives of the chain rule, mean value theorem, and Taylor’s formula among others which become highly useful in the analysis of generalized Banach space problems and their solutions in Rn.
Sentient Arithmetic and Godel's Incompleteness TheoremsKannan Nambiar
For me, there is only one logic that we rational human beings are able to accept and appreciate, and that is the mathematical logic of ZF theory. But in the last century we found that ZF theory is not in a position to provide all that we want, and went in search of a new mode of thinking and got one which we called meta mathematics. My question is: if we can put the unambiguous logic of ZF theory on paper, why can't we do the same with meta mathematics. This paper is my feeble attempt in that direction.
ANURAG TYAGI CLASSES (ATC) is an organisation destined to orient students into correct path to achieve
success in IIT-JEE, AIEEE, PMT, CBSE & ICSE board classes. The organisation is run by a competitive staff comprising of Ex-IITians. Our goal at ATC is to create an environment that inspires students to recognise and explore their own potentials and build up confidence in themselves.ATC was founded by Mr. ANURAG TYAGI on 19 march, 2001.
MEET US AT:
www.anuragtyagiclasses.com
Some Characterizations of Riesz Valued Sequence Spaces generated by an Order ...UniversitasGadjahMada
In this paper, we introduce an order φ-function on a Riesz space. Further, we construct a Riesz valued sequence spaces by using the φ-function and obtain the condition to characterize these spaces.
From the course of mechanics of solids (CE-205).
the slide contains:
1. what is beam and its types
2.whatis shearing stress and bending moment (video including)
3.causes of shearing stress.
4. types of shearing stress.
5. numerical example.
Cosmetic shop management system project report.pdfKamal Acharya
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.
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2. Invented by Joseph Fourier
DEFINITION:
A Fourier series is an expansion of a periodic function
f(x) in terms of an infinite sum of sines and cosines.
Fourier series make use of the orthogonality
relationships of the sine and cosine functions
The computation and study of Fourier series is known
as harmonic analysis and is extremely useful as a way to
break up an arbitrary periodic function into a set of
simple terms that can be plugged in, solved individually,
and then recombined to obtain the solution to the original
problem or an approximation to it to whatever accuracy
is desired or practical. Examples of successive
approximations to common functions using Fourier
series are illustrated above.
4. Fourier series can generally be written as:
Where,
n=1, 2, 3, .... Note that the coefficient of the constant term a_0 has been written in a special form
compared to the general form for a generalized Fourier series in order to preserve symmetry with
the definitions of a_n and b_n.
5. Fourier series for even and odd function
If a function is even so that f(x)=f(-x), then
f(x)sin(nx) is odd. (This follows since sin(nx) is
odd and an even function times an odd function
is an odd function.) Therefore, b_n=0 for all n.
Similarly, if a function is odd so that f(x)=-f(-x), then
f(x)cos(nx) is odd. (This follows since cos(nx) is
even and an even function times an odd function is
an odd function.) Therefore, a_n=0 for all n.
Even Function Odd Function
6. EXAMPLE:
Question : find the Fourier series of f(x)= 𝑥2
+ 𝓍
Answer: The Fourier series of f(x) is given by
