Mid point
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The document describes the midpoint line drawing algorithm. It begins with an overview of the algorithm and its incremental nature. It then provides the key equations used in the algorithm to calculate the midpoint and determine if it is above or below the line. Based on these calculations, it decides whether the next pixel will have a y-value of the current or next y-coordinate. Worked examples are provided to demonstrate calculating the intermediate pixels between two given points using the algorithm.
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monotonicity thoery & solved & execise Module-4.pdf
In this chapter, we shall study the nature of a
function which is governed by the sign of its derivative. If the graph of a function is in upward going direction or in downward coming direction then it is called as monotonic function, and this property of the function is called Monotonicity. If a function is defined in any interval, and if in some part of the interval, graph moves upwards
A function is said to be monotonic function in a domain if it is either monotonic increasing or monotonic decreasing in that domain.
NOTE : If x < x f(x ) < f(x ) x , x D, then
and in the remaining part moves downward then
1 2 1
2 1 2
function is not monotonic in that interval.
f(x) is called strictly increasing in domain D.
These are of two types –
2.1 Monotonic Increasing :
A function f(x) defined in a domain D is said to be monotonic increasing function if the value of f(x) does not decrease (increase) by increasing (decreasing) the value of x or
We can say that the value of f(x) should increase (decrease) or remain equal by increasing
(Decreasing) the value of x.
Similarly if x1 < x2 f(x1) > f(x2), x1, x2 D then it is called strictly decreasing in domain D.
R
If Tor x1 x2 f(x1) f(x2 )
, x1 , x2 D
or SR x1 x2 f(x1) f(x2 ) , x1 , x2 D
2.2 Monotonic Decreasing :
A function f(x) defined in a domain D is said to be monotonic decreasing function if the value of f(x) does not increase (decrease) by increasing (decreasing) the value of x or
We can say that the value of f(x) should decrease (increase) or remain equal by increasing
(Decreasing) the value of x.
For Example
(i) f(x) = ex is a monotonic increasing function where as g(x) = 1/x is monotonic decreasing function.
(ii) f(x) = x2 and g(x) = | x | are monotonic increasing for x > 0 and monotonic decreasing for x < 0. In general they are not monotonic functions.
(iii) Sin x, cos x are not monotonic function whereas tan x, cot x are monotonic.
(i) At a Point : A function f(x) is said to be monotonic increasing (decreasing) at a point x = a of its domain if it is monotonic increasing
R
If Tor x1 x2 f(x1) f(x2 )
, x1 , x2 D
(decreasing) in the interval (a – h, a + h) where h is a small positive number. Hence we may
or SR x1 x2 f(x1) f(x2 ) , x1 , x2 D
observe that if f(x) is monotonic increasing at x = a, then at this point tangent to its graph will make an acute angle with the x–axis where as if the function is monotonic decreasing these tangent will make
an obtuse angle with x–axis. Consequently f' (a) will be positive or negative according as f(x) is monotonic increasing or decreasing at x = a.
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Ex. Function f(x) = x2 is not a monotonic function in the interval [–1, 1] because
f' (x) > 0, when x = 1/2
Ex. Function f(x) = sin2x + cos2x is constant
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2. Vertical stretches and compressions which multiply the y-values of a graph by a constant factor c.
3. Horizontal translations which shift a graph left or right by adding or subtracting a constant c to the x-values.
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A note on the density of Gumbel-softmax
This note explicates some details of the discussion
given in Appendix B of E. Jang, S. Gu, and B. Poole.
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ICLR, 2017.
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The document summarizes Bresenham's line drawing algorithm. It derives the equations for calculating the next pixel position when drawing a line on a digital display. It considers cases where the slope is less than or equal to 1 and greater than 1. For each case, it calculates the distance from intersection points to pixel positions and derives the decision parameter equation to determine the next pixel.
Chapter 3 Output Primitives
This includes different line drawing algorithms,circle,ellipse generating algorithms, filled area primitives,flood fill ,boundary fill algorithms,raster scan fill approaches.
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The document provides a step-by-step guide for finding the vertex of parabolic functions by completing the square. It gives two examples, finding that the vertex of f(x)=x^2 -4x+3 is (2,-1) and the vertex of f(x)=-2x^2 -2x+1 is (-1/2,-1). Completing the square involves factoring the quadratic term and rearranging constants to put the function in vertex form f(x)=a(x-h)^2 + k, where (h,k) gives the vertex coordinates.
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Program to create a cylinder and Parallelepiped by extruding Circle and Quadrilateral respectively. Allow the user to specify the circle and quadrilateral.
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The document discusses computer graphics output primitives and line drawing algorithms. It describes points, lines, polygons and other basic geometric structures used to describe scenes in graphics. It then explains two common line drawing algorithms - the Digital Differential Analyzer (DDA) algorithm and Bresenham's line drawing algorithm. Bresenham's algorithm uses only integer calculations to efficiently rasterize lines and is often used in computer graphics.
Difference quotient algebra
The document discusses the difference quotient formula for calculating the slope between two points (x1,y1) and (x2,y2) on a function y=f(x). It shows that the slope m is equal to (f(x+h)-f(x))/h, where h is the difference between x1 and x2. This "difference quotient" formula allows slopes to be calculated from the values of a function at two nearby points. Examples are given of simplifying the difference quotient for quadratic and rational functions.
Lesson20 Tangent Planes Slides+Notes
The document is notes for a lesson on tangent planes. It provides definitions of tangent lines and planes, formulas for finding equations of tangent lines and planes, and examples of applying these concepts. Specifically, it defines that the tangent plane to a function z=f(x,y) through the point (x0,y0,z0) has normal vector (f1(x0,y0), f2(x0,y0),-1) and equation f1(x0,y0)(x-x0) + f2(x0,y0)(y-y0) - (z-z0) = 0 or z = f(x0,y0) +
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The document provides examples of finding the vertex of parabolic functions by completing the square. It shows working through examples where the coefficient of x^2 is 1 and where it is not 1. In both cases, the process involves separating terms containing x, squaring half the coefficient of x, and factorizing the resulting expression to put it in the vertex form of f(x)=(x-h)^2+k, where (h,k) gives the vertex.
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在线办理(salfor毕业证书)索尔福德大学毕业证毕业完成信一模一样
学校原件一模一样【微信：741003700 】《(salfor毕业证书)索尔福德大学毕业证》【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
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◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
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8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
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Equivariant neural networks and representation theory
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
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The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
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https://www.etran.rs/2024/en/home-english/原版制作(carleton毕业证书)卡尔顿大学毕业证硕士文凭原版一模一样
原版纸张【微信：741003700 】【(carleton毕业证书)卡尔顿大学毕业证】【微信：741003700 】学位证，留信认证（真实可查，永久存档）offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原海外各大学 Bachelor Diploma degree, Master Degree Diploma
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
ESR spectroscopy in liquid food and beverages.pptx
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
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Mid point
- 1. Presentation on: Mid- point line drawing algorithm SUBMITTED BY: Juhi Kumari
- 2. Midpoint algorithm is an incremental algorithm. Candidate Pixels Current Pixel ( xk, yk) Midpoint Line Coordinates of Midpoint = ( xk+1, yk+(1/2) ) ( xk+1, yk) ( xk+1, yk+1)
- 3. Equation of the line: y-y1/x-x1=y2-y1/x2-x1 ----(1) Let w = x2 x1, and h = y2 y1 By using equation (1) y-y1/x-x1=h/w Then, h (x x1) w (y y1) = 0. (h, w , x1, y1 are all integers). In other words, every point (x, y) on the line satisfies the equation F(x, y) =0, where F(x, y) = h (x x1) w (y y1). A(x1,y1) B(x2,y2)
- 4. Decision criteria F(MP) = Fk =F(xk+1, yk+ ½) ------ (Notation) If Fk < 0 : The midpoint is above the line. So the next pixel is (xk+1, yk). If Fk 0 : The midpoint is below or on the line. So the next pixel is (xk+1, yk+1).
- 5. Choice of the next pixel Midpoint Below Line Next pixel = (xk+1, yk+1) Fk > 0 yk+1 = yk+1 Midpoint Above Line Next pixel = (xk+1, yk) Fk < 0 yk+1 = yk
- 6. Fk = F(xk+1, yk+ ½) By using equation, Fk = h (x x1) w (y y1) Fk = h (xk+1 x1) w (yk+½ y1)------(1) But, Fk+1 = ( xk+1+1, yk+1+(1/2) ) =h(xk+1+1-x1)-w( yk+1+1/2-y1) = h(xk+2-x1)-w( yk+1+1/2-y1) Now, Fk+1 -Fk =h w (yk+1 yk) = Fk + h w (yk+1 yk)
- 7. So, Fk< 0 : yk+1 = yk. Hence, Fk+1 = Fk + h . Fk 0 : yk+1 = yk+1. Hence, Fk+1 = Fk + h w. Now, putting k=0 in equation (1), F0 = h w/2.
- 8. int h = y2-y1; int w = x2-x1; float F=h-w/2; int x=x1, y=y1; for (x=x1; x<=x2; x++) { setPixel(x, y); if(F < 0) F = F+h; else{ F = F+h-w; y++; } }
- 9. QUES: Find the intermediate pixels of the point (1,2) and (6,6) using mid point line drawing algorithm? ANS: Given, x1=1,y1=2 x2=6,y2=6 now, h=y2-y1=6-2=4 w=x2-x1=6-1=5 f=(2*h)-w=(2*4)-5=3 Points are: (1,2); (2,3); (3,4); (4,5); (5,6); (6,6)