The document discusses different methods of polynomial interpolation, including Lagrange interpolation, Newton's divided differences, and spline interpolation. Lagrange interpolation uses polynomials of varying degrees to find intermediate values between known data points. Newton's divided differences uses a recursive method to find polynomial coefficients for interpolation. Spline interpolation involves fitting piecewise polynomials between data points, with linear splines using first-order polynomials between points and cubic splines using third-order polynomials. Examples are provided for each method to demonstrate how to derive the interpolating polynomials and find outputs at given x-values.
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
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.
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Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
HEAP SORT ILLUSTRATED WITH HEAPIFY, BUILD HEAP FOR DYNAMIC ARRAYS.
Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to the selection sort where we first find the minimum element and place the minimum element at the beginning. Repeat the same process for the remaining elements.
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.
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
6. 1st Order LAGRANGE POLYNOMIAL
INTERPOLATION
Example: Find y at x = 2.2
# Find 1st order Lagrange Polynomial Equation
# Plug in data and Simplify, and Validate
# Plug in given data to newly found function
f(x) =
𝒙−𝒙𝟐
𝒙𝟏−𝒙𝟐
y1 +
𝒙−𝒙𝟏
𝒙𝟐−𝒙𝟏
y2
*** Short Cut
X Y
2 5
3 10
7. 2nd Order LAGRANGE POLYNOMIAL
INTERPOLATION
Example: Find y at x = 2.5
n=3 thus 2nd order Lagrange
# Find General order Lagrange Polynomial Equation
# Plug in data and Simplify; then Validate
# Plug in given data to newly found function
f(x) =
(𝒙−𝒙𝟐)(𝒙−𝒙𝟑)
𝒙𝟏
−𝒙𝟐
(𝒙𝟏
−𝒙𝟑
)
y1 +
(𝒙−𝒙𝟏)(𝒙−𝒙𝟑)
𝒙𝟐
−𝒙𝟏
(𝒙𝟐
−𝒙𝟑
)
y2 +
(𝒙−𝒙𝟏)(𝒙−𝒙𝟐)
𝒙𝟑
−𝒙𝟏
(𝒙𝟑
−𝒙𝟐
)
y3
X Y
1 4
3 8
7 10
8. 2nd Order LAGRANGE POLYNOMIAL
INTERPOLATION
Example: Find y at x = 2.5
n=3 thus 2nd order Lagrange
f(x) =
(𝒙−𝒙𝟐
)(𝒙−𝒙𝟑
)
𝒙𝟏
−𝒙𝟐
(𝒙𝟏
−𝒙𝟑
)
y1 +
(𝒙−𝒙𝟏
)(𝒙−𝒙𝟑
)
𝒙𝟐
−𝒙𝟏
(𝒙𝟐
−𝒙𝟑
)
y2 +
(𝒙−𝒙𝟏
)(𝒙−𝒙𝟐
)
𝒙𝟑
−𝒙𝟏
(𝒙𝟑
−𝒙𝟐
)
y3
= - 0.25 x2 + 3x + 1.25
if x = 2.5, then y should be 7.1875.
X Y
1 4
3 8
7 10
9. 3rd Order LAGRANGE POLYNOMIAL
INTERPOLATION
Example: Find y at x = 3
n=4 thus 3rd order Lagrange
f(x) =
(𝒙−𝒙𝟐
)(𝒙−𝒙𝟑
)(𝒙−𝒙𝟒
)
𝒙𝟏
−𝒙𝟐
(𝒙𝟏
−𝒙𝟑
)(𝒙𝟏
−𝒙𝟒
)
y1 +
(𝒙−𝒙𝟏
)(𝒙−𝒙𝟑
)(𝒙−𝒙𝟒
)
𝒙𝟐
−𝒙𝟏
(𝒙𝟐
−𝒙𝟑
)(𝒙𝟐
−𝒙𝟒
)
y2 +
(𝒙−𝒙𝟏
)(𝒙−𝒙𝟐
)(𝒙−𝒙𝟒
)
𝒙𝟑
−𝒙𝟏
(𝒙𝟑
−𝒙𝟐
)(𝒙𝟑
−𝒙𝟒
)
y3 +
(𝒙−𝒙𝟏
)(𝒙−𝒙𝟐
)(𝒙−𝒙𝟑
)
𝒙𝟒
−𝒙𝟏
(𝒙𝟒
−𝒙𝟐
)(𝒙𝟒
−𝒙𝟑
)
y4
= 0.25 x3 – 1.5833x2 + 5x + 0.333
if x = 3, then y should be 7.833.
X Y
1 4
2 6
4 11
5 17
11. NEWTON’S DIVIDED DIFFERENCE &
POLYNOMIAL INTERPOLATION
DIVIDED DIFFERENCE INTERPOLATION Numerical Interpolation
method to find the coefficients of a curve fitting polynomial
Newton’s polynomial polynomials that we use to interpolate for a specified
set of data. Advantage: faster, recursive, better
18. GLOBAL VS LOCAL INTERPOLATION
Interpolation used to find continuous (& ideally smooth) functions from discrete data
points Interpolating function f(x)
Discrete data points
Interpolation Process:
Given data set
Fit interpolating function for the given data set
Use newly found function to find the output for any input (within domain)
Y
X
19. GLOBAL VS LOCAL INTERPOLATION
GLOBAL INTERPOLATION LOCAL INTERPOLATION
Uses all supplied data to create the interpolating
function; Single/Higher order polynomial
Uses only a subset of all supplied data points
Consists of lower order polynomials
E.g. Lagrange Polynomials, Divided Difference E.g. Spline Interpolation
Must use all data; Always gives the same answer Local Interpolation can use all or little of our supplied
data (must all be continuous and connecting)
Problem: Increase polynomial order = Increase error
at edges of our equal distance input points
Y
X
Global Polynomial
364th Order
Local Polynomial
1st or 2nd
Order
21. TYPES OF SPLINE INTERPOLATIONS
1ST Order – Linear Spline 2nd Order – Cubic Spline
More widely used
Y
Y1
Y0, Y2
X0 X1 X2 X
Y
Y1
Y0, Y2
X0 X1 X2 X
22. LINEAR SPLINE INTERPOLATIONS
1ST Order – Linear Spline
These often lead to knots/sharp changes in our function which
is un-ideal as we want to smooth continuous functions
P2(x)
P1(x) We assume our function to be linear
** How de we go about finding P1(x) and a point along it?
EACH SEGMENT IS SIMPLY A STRAIGHT LINE EQUATION
Gen. Equation of Line: y = mx + b
P1(x) = Y1 +
𝑌2
−𝑌1
𝑋2
−𝑋1
(X – X1)
P2(x) = Y2 + 𝑌3
−𝑌2
𝑋3
−𝑋2
(X – X2)
Y
Y1
Y0, Y2
X0 X1 X2 X
23. LINEAR SPLINE INTERPOLATIONS
Given
Find the necessary interpolating functions
Find the outputs at x = 2, 5 and 10
P2(x)
P1(x)
Gen. Equation of Line: y = mx + b
P1(x) = 2 + 8−2
6−1
(X – 1) = 1.2x + 0.8
1 ≤ 𝑥 ≤ 6
P2(x) = 10 +
14−10
12−9 (X – 9) = 4
3
𝑥 - 2 @ x=2, use P1(x) y=3.2
9 ≤ 𝑥 ≤ 12 @ x=5, use P1(x) y=6.8
@ x=10, use P2(x) y=11.3333
X Y
1 2
6 8
7 6
9 10
12 14
20 41
Y
X
24. QUADRATIC SPLINE INTERPOLATIONS
Given
Find the necessary interpolating functions
Find the outputs at x = 2, 4 and 7
P3(x)
*** 3n equations P2(x)
3 splines = 9 unknowns P1(x)
Write out General polynomials
Identify unknowns
Solve unknowns
Plug in X-inputs
X Y
1 2
3 3
5 9
8 10
Y
X
25. QUADRATIC SPLINE INTERPOLATIONS
X Y
1 2
3 3
5 9
8 10
Y
X
P2(x)
P1(x)
P3(x)
(1) Polynomials to find:
P1(x) = a1 X2 + b1 X + c1
P2(x) = a2 X2 + b2 X + c2
P3(x) = a3 X2 + b3 X + c3
(2) 9 unknowns a1, b1, c1, a2, b2, c2, a3, b3, c3