THE NUMERICAL SOLUTION OF INTEGRAL-ALGEBRAIC EQUATIONS OF INDEX 1 BY POLYNOMIAL SPLINE COLL

算术平均牛顿法的英文Arithmetic-Geometric Mean Newton's Method.The arithmetic-geometric mean (AGM) Newton's method is an iterative algorithm used in numerical analysis to approximate the solution of equations, particularly those involving transcendental functions. This method is avariant of the classical Newton's method, which uses the tangent line to the function at a given point to approximate the root of the function. The AGM Newton's method incorporates the arithmetic-geometric mean (AGM) iteration, which is itself a fast converging method for computing the square root of a number.Background on Newton's Method:Newton's method is based on the Taylor series expansion of a function. Given a function f(x) and its derivativef'(x), the method starts with an initial guess x0 and iteratively updates the approximation using the formula:x_{n+1} = x_n f(x_n) / f'(x_n)。

Modeling Illumination Variation with Spherical Harmonics
Ravi Ramamoorthi Columbia Universi
Introduction
Illumination can have a significant impact on the appearance of surfaces, as the patterns of shading, specularities and shadows change. For instance, some images of a face under different lighting conditions are shown in figure 1. Differences in lighting can often play a much greater role in image variability of human faces than differences between individual people. Lighting designers in movies can often set the mood of a scene with carefully chosen lighting. To achieve a sinister effect, for instance, one can use illumination from below the subject—a sharp contrast to most natural indoor or outdoor scenes where the dominant light sources are above the person. Characterizing the variability in appearance with lighting is a fundamental problem in many areas of computer vision, face modeling, and computer graphics. One of the great challenges of computer vision is to produce systems that can work in uncontrolled environments. To be robust, recognition systems must be able to work oudoors in a lighting-insensitive manner. In computer graphics, the challenge is to be able to efficiently create the visual appearance of a scene under realistic, possibly changing, illumination. At first glance, modeling the variation with lighting may seem intractable. For instance, a video projector can illuminate an object like a face with essentially any pattern. In this chapter, we will stay away from such extreme examples, making a set of assumptions that are approximately true in many common situations. One assumption we make is that illumination is distant. By this, we mean that the direction to, and intensity of, the light sources is approximately the same throughout the region of interest. This explicitly rules out cases like slide projectors. This is a reasonably good approximation in outdoor scenes, where the sky can be assumed to be far away. It is also fairly accurate in many indoor environments, where the light sources can be considered much further away relative to the size of the object. Even under the assumption of distant lighting, the variations may seem intractable. The illumination can come from any incident direction, and can be composed of multiple illuminants including localized light sources like sunlight and broad area distributions like skylight. In the general case, we would need to model the intensity from each of infinitely many incident lighting directions. Thus, the space we are dealing with appears to be infinite-dimensional. By contrast, a number of other causes of appearance variation are low-dimensional. For instance, appearance varies with viewing direction as well. However, unlike lighting, there can only be a single view direction. In general, variation because of pose and translation can be described using only six degrees of freedom. Given the daunting nature of this problem, it is not surprising that most previous analytic models have been restricted to the case of a single directional (distant) light source, usually without considering shadows. In computer graphics, this model is sufficient (but not necessarily efficient) for numerical Monte Carlo simulation of appearance, since one can simply treat multiple light sources separately and add up the results. In computer vision, such models can be reasonable approximations under some circumstances, such as controlled laboratory conditions. However, they do not suffice for modeling illumination variation in uncontrolled conditions like the outdoors. Fortunately, there is an obvious empirical method of analyzing illumination variation. One can simply record a number of images of an object under light sources from all different directions. In practice, this usually corresponds to moving a light source in a sphere around the object or person, while keeping camera and pose fixed. Because of the linearity of light transport—the image under multiple lights is the sum of that under the individual sources—an image under arbitrary distant illumination can be written as a linear combination of these source images. This observation in itself provides an attractive direct approach for relighting in computer graphics. Furthermore, instead of using the source images, we can try to find linear combinations or basis images that best explain the

微积分介值定理的英文The Intermediate Value Theorem in CalculusCalculus, a branch of mathematics that has revolutionized the way we understand the world around us, is a vast and intricate subject that encompasses numerous theorems and principles. One such fundamental theorem is the Intermediate Value Theorem, which plays a crucial role in understanding the behavior of continuous functions.The Intermediate Value Theorem, also known as the Bolzano Theorem, states that if a continuous function takes on two different values, then it must also take on all values in between those two values. In other words, if a function is continuous on a closed interval and takes on two different values at the endpoints of that interval, then it must also take on every value in between those two endpoint values.To understand this theorem more clearly, let's consider a simple example. Imagine a function f(x) that represents the height of a mountain as a function of the distance x from the base. If the function f(x) is continuous and the mountain has a peak, then theIntermediate Value Theorem tells us that the function must take on every height value between the base and the peak.Mathematically, the Intermediate Value Theorem can be stated as follows: Let f(x) be a continuous function on a closed interval [a, b]. If f(a) and f(b) have opposite signs, then there exists a point c in the interval (a, b) such that f(c) = 0.The proof of the Intermediate Value Theorem is based on the properties of continuous functions and the completeness of the real number system. The key idea is that if a function changes sign on a closed interval, then it must pass through the value zero somewhere in that interval.One important application of the Intermediate Value Theorem is in the context of finding roots of equations. If a continuous function f(x) changes sign on a closed interval [a, b], then the Intermediate Value Theorem guarantees that there is at least one root (a value of x where f(x) = 0) within that interval. This is a powerful tool in numerical analysis and the study of nonlinear equations.Another application of the Intermediate Value Theorem is in the study of optimization problems. When maximizing or minimizing a continuous function on a closed interval, the Intermediate Value Theorem can be used to establish the existence of a maximum orminimum value within that interval.The Intermediate Value Theorem is also closely related to the concept of connectedness in topology. If a function is continuous on a closed interval, then the image of that interval under the function is a connected set. This means that the function "connects" the values at the endpoints of the interval, without any "gaps" in between.In addition to its theoretical importance, the Intermediate Value Theorem has practical applications in various fields, such as economics, biology, and physics. For example, in economics, the theorem can be used to show the existence of equilibrium prices in a market, where supply and demand curves intersect.In conclusion, the Intermediate Value Theorem is a fundamental result in calculus that has far-reaching implications in both theory and practice. Its ability to guarantee the existence of values between two extremes has made it an indispensable tool in the study of continuous functions and the analysis of complex systems. Understanding and applying this theorem is a crucial step in mastering the powerful concepts of calculus.。

β α a It (a It y (t)) α+β β +α β α = a It y (t) = a It y (t) = a It (a It y (t)) .
(6)
In the general case the Riemann-Liouville fractional derivatives do not commute
β α+β β α α (a Dt (a Dt y ))(t) = (a Dt )(t) = (a Dt (a Dt y ))(t) .
(7)
Extending our considerations, the integer operator commutes with the fractional operator
β α−β α (a Dt (a It y ))(t) = (a Dt )(t) ,
(9)
but they do not commute in the opposite way. We turn our attention to the composition rule in two following ways. First of all we found in literature [10,11] the fact, that authors neglect the general property of fractional derivatives given by eqn. (7). Regarding to solution of fractional differential equations we will apply above properties in the next sections.
α y )(t) = (a It