ISOPERIMETRIC PROBLEMS OF THE CALCULUS OF VARIATIONS WITH FRACTIONAL DERIVATIVES
微积分calculus英文单词
微积分英语单词Absolute convergence ：绝对收敛Absolute extreme values ：绝对极值Absolute maximum and minimum ：绝对极大与极小Absolute value ：绝对值Absolute value function ：绝对值函数Acceleration ：加速度Antiderivative ：反导数Approximate integration ：近似积分Approximation ：逼近法Arc length ：弧长Area ：面积Asymptote ：渐近线Average speed ：平均速率Average velocity ：平均速度Axes, coordinate ：坐标轴Axes of ellipse ：椭圆之轴at a point ：在一点处之连续性as the slope of a tangent ：导数看成切线之斜率by differentials ：用微分逼近between curves ：曲线间之面积Binomial series ：二项级数Cartesian coordinates ：笛卡儿坐标一般指直角坐标Cartesian coordinates system ：笛卡儿坐标系Cauch’s Mean Value Theorem ：柯西均值定理Chain Rule ：连锁律Change of variables ：变数变换Circle ：圆Circular cylinder ：圆柱Closed interval ：封闭区间Coefficient ：系数Composition of function ：函数之合成Compound interest ：复利Concavity ：凹性Conchoid ：蚌线Cone ：圆锥Constant function ：常数函数Constant of integration ：积分常数Continuity ：连续性Continuous function ：连续函数Convergence ：收敛Coordinate ：s ：坐标Cartesian ：笛卡儿坐标cylindrical ：柱面坐标Coordinate axes ：坐标轴Coordinate planes ：坐标平面Cosine function ：余弦函数Critical point ：临界点Cubic function ：三次函数Curve ：曲线Cylinder ：圆柱Cylindrical Coordinates ：圆柱坐标Distance ：距离Divergence ：发散Domain ：定义域Dot product ：点积Double integral ：二重积分Decreasing function ：递减函数Decreasing sequence ：递减数列Definite integral ：定积分Degree of a polynomial ：多项式之次数Density ：密度Derivative ：导数Determinant ：行列式Differentiable function ：可导函数Differential ：微分Differential equation ：微分方程Differentiation ：求导法Directional derivatives ：方向导数Discontinuity ：不连续性Disk method ：圆盘法domain of ：导数之定义域differential ：微分学Ellipse ：椭圆Ellipsoid ：椭圆体Epicycloid ：外摆线Equation ：方程式Even function ：偶函数Expected Valued ：期望值Exponential Function ：指数函数Exponents , laws of ：指数率Extreme value ：极值Extreme Value Theorem ：极值定理Factorial ：阶乘First Derivative Test ：一阶导数试验法First octant ：第一卦限Focus ：焦点Fractions ：分式Function ：函数Fundamental Theorem of Calculus ：微积分基本定理from the left ：左连续from the right ：右连续Geometric series ：几何级数Gradient ：梯度Graph ：图形Green Formula ：格林公式Half-angle formulas ：半角公式Harmonic series ：调和级数Helix ：螺旋线Higher Derivative ：高阶导数Horizontal asymptote ：水平渐近线Horizontal line ：水平线Hyperbola ：双曲线Hyperboloid ：双曲面horizontal ：水平渐近线Implicit differentiation ：隐求导法Implicit function ：隐函数Improper integral ：瑕积分Increasing/Decreasing Test ：递增或递减试验法Increment ：增量Increasing Function ：增函数Indefinite integral ：不定积分Independent variable ：自变数Indeterminate from ：不定型Inequality ：不等式Infinite point ：无穷极限Infinite series ：无穷级数Inflection point ：反曲点Instantaneous velocity ：瞬时速度Integer ：整数Integral ：积分Integrand ：被积分式Integration ：积分Integration by part ：分部积分法Intercepts ：截距Intermediate value of Theorem ：中间值定理Interval ：区间Inverse function ：反函数Inverse trigonometric function ：反三角函数Iterated integral ：逐次积分integral ：积分学implicit ：隐求导法Laplace transform ：Leplace 变换Law of Cosines ：余弦定理Least upper bound ：最小上界Left-hand derivative ：左导数Left-hand limit ：左极限Lemniscate ：双钮线Length ：长度Level curve ：等高线L'Hospital's rule ：洛必达法则Limacon ：蚶线Limit ：极限Linear approximation：线性近似Linear equation ：线性方程式Linear function ：线性函数Linearity ：线性Linearization ：线性化Line in the plane ：平面上之直线Line in space ：空间之直线Lobachevski geometry ：罗巴切夫斯基几何Local extremum ：局部极值Local maximum and minimum ：局部极大值与极小值Logarithm ：对数Logarithmic function ：对数函数linear ：线性逼近法Maximum and minimum values ：极大与极小值Mean Value Theorem ：均值定理Multiple integrals ：重积分Multiplier ：乘子Natural exponential function ：自然指数函数Natural logarithm function ：自然对数函数Natural number ：自然数Normal line ：法线Normal vector ：法向量Number ：数of a function ：函数之连续性on an interval ：在区间之连续性Octant ：卦限Odd function ：奇函数One-sided limit ：单边极限Open interval ：开区间Optimization problems ：最佳化问题Order ：阶Ordinary differential equation ：常微分方程 Origin ：原点Orthogonal ：正交的Parabola ：拋物线Parabolic cylinder ：抛物柱面Paraboloid ：抛物面Parallelepiped ：平行六面体Parallel lines ：并行线Parameter ：参数Partial derivative ：偏导数Partial differential equation ：偏微分方程 Partial fractions ：部分分式Partial integration ：部分积分Partiton ：分割Period ：周期Periodic function ：周期函数Perpendicular lines ：垂直线Piecewise defined function ：分段定义函数 Plane ：平面Point of inflection ：反曲点Polar axis ：极轴Polar coordinate ：极坐标Polar equation ：极方程式Pole ：极点Polynomial ：多项式Positive angle ：正角Point-slope form ：点斜式Power function ：幂函数Product ：积polar ：极坐标partial ：偏导数partial ：偏微分方程partial ：偏微分法Quadrant ：象限Quotient Law of limit ：极限的商定律Quotient Rule ：商定律rectangular ：直角坐标Radius of convergence ：收敛半径Range of a function ：函数的值域Rate of change ：变化率Rational function ：有理函数Rationalizing substitution ：有理代换法Rationalizing substitution ：有理代换法Rational number ：有理数Real number ：实数Rectangular coordinates ：直角坐标Rectangular coordinate system ：直角坐标系Relative maximum and minimum ：相对极大值与极小值Revenue function ：收入函数Revolution, solid of ：旋转体Revolution, surface of ：旋转曲面Riemann Sum ：黎曼和Riemannian geometry ：黎曼几何Right-hand derivative ：右导数Right-hand limit ：右极限Root ：根Saddle point ：鞍点Scalar ：纯量Secant line ：割线Second derivative ：二阶导数Second Derivative Test ：二阶导数试验法Second partial derivative ：二阶偏导数Sector ：扇形Sequence ：数列Series ：级数Set ：集合Shell method ：剥壳法Sine function ：正弦函数Singularity ：奇点Slant asymptote ：斜渐近线Slope ：斜率Slope-intercept equation of a line ：直线的斜截式Smooth curve ：平滑曲线Smooth surface ：平滑曲面Solid of revolution ：旋转体Space ：空间Speed ：速率Spherical coordinates ：球面坐标Squeeze Theorem ：夹挤定理Step function ：阶梯函数Strictly decreasing ：严格递减Strictly increasing ：严格递增Sum ：和Surface ：曲面Surface integral ：面积分Surface of revolution ：旋转曲面Symmetry ：对称slant ：斜渐近线spherical ：球面坐标Tangent function ：正切函数Tangent line ：切线Tangent plane ：切平面Tangent vector ：切向量Total differential ：全微分Trigonometric function ：三角函数Trigonometric integrals ：三角积分Trigonometric substitutions ：三角代换法Tripe integrals ：三重积分term by term ：逐项求导法under a curve ：曲线下方之面积vertical ：垂直渐近线Value of function ：函数值Variable ：变数Vector ：向量Velocity ：速度Vertical asymptote ：垂直渐近线Volume ：体积X-axis ：x 轴x-coordinate ：x 坐标x-intercept ：x 截距Zero vector ：函数的零点Zeros of a polynomial ：多项式的零点。
Calculus I
Calculus ICalculus, also known as mathematical analysis, is a branch of mathematics that deals with the study of rates of change and how things change over time. It is a fundamental mathematical tool that has become essential in many fields such as physics, engineering, economics, and biology. In this essay, we will explore Calculus I, which is the introductory course for Calculus.The study of Calculus is divided into two main branches: differential calculus and integral calculus. Differential calculus is concerned with the study of rates of change, while integral calculus is concerned with the study of accumulation. Calculus I focuses on the fundamental concepts of differential calculus.One of the key ideas in Calculus I is the concept of limit. A limit is the value that a function approaches as the independent variable approaches a certain value. Limits are an essential tool for studying the behavior of functions, especially at points where the function may not be defined.Another important concept in Calculus I is the derivative. The derivative of a function is the rate of change of the function at a particular point. It is defined as the limit of the difference quotient as the change in the independent variable approaches zero. The derivative is a fundamental concept in Calculus and is used extensively in many fields, including physics, engineering, and economics.The derivative has many important properties, including the power rule, product rule, quotient rule, and chain rule. These rules allowus to find the derivative of complicated functions quickly and efficiently.The derivative also has many applications, including optimization problems and finding the location of maximum and minimum values of a function. For example, in economics, the derivative is used to find the marginal cost and marginal revenue of a company. In physics, the derivative is used to find the instantaneous velocity and acceleration of an object.Another important concept in Calculus I is the notion of differentiation. Differentiation is the process of finding the derivative of a function. It is an integral part of Calculus and is used extensively in many fields.One of the most important applications of differentiation is in the study of optimization problems. Optimization problems involve finding the maximum or minimum value of a function subject to certain constraints. For example, in economics, firms try to maximize their profits subject to certain constraints, such as the cost of production.Integration is the second branch of Calculus, and it deals with finding the area under a curve. Integration is the inverse of differentiation, and it is used extensively in many fields, including physics and engineering.One of the most important applications of integration is in the study of volumes and areas. For example, in physics, the volume of a solid can be found by integrating the area under the curve of itscross-section. In engineering, the area of an irregular shape can be found by integrating the area under the curve of its boundary.Calculus I also covers important topics such as limits, continuity, and trigonometric functions. Limits are used extensively in Calculus to study the behavior of functions. Continuity is a fundamental concept in Calculus that ensures that a function is well-behaved and has no abrupt changes.Trigonometric functions are essential in Calculus because they are used extensively in the study of differential equations, which are equations that involve derivatives. Differential equations are used to model many real-world phenomena, such as the growth of a population and the spread of diseases.In conclusion, Calculus I is an essential course for any student studying mathematics, physics, engineering, or economics. It provides a solid foundation for more advanced courses in Calculus and other fields. The concepts of differential calculus, such as limits, derivatives, and differentiation, are fundamental in the study of many real-world problems. The concepts covered in Calculus I, such as optimization and integration, have many applications in numerous fields and are essential for solving problems in many areas of science and engineering.In addition to the topics mentioned above, Calculus I also covers related rates, which are useful in real-world scenarios where things are changing at different rates. For example, if you are filling a pool with water and you want to know how fast the water level is rising, you would use related rates. This involves finding the relationship between the rates of change of different variables and using this relationship todetermine one rate when the other rate is known.Another important concept in Calculus I is the Mean Value Theorem. This theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists a point in the interval where the derivative is equal to the average rate of change of the function over the interval. This theorem has applications in many areas, including economics, where it is used to prove the existence of equilibrium prices.Calculus I also covers curve sketching, which involves studying the behavior of a function as it approaches zero and infinity, finding its intercepts, and determining where it is increasing or decreasing. This is important in many fields as it allows us to understand the behavior of functions and predict their future values.One of the most important applications of Calculus I is in physics, where it is used extensively in studying motion. The concepts of calculus are used to determine the velocity, acceleration, and position of an object at any given point in time. Understanding these concepts is essential in fields such as aerospace engineering, where the motion of objects in space is critical to the success of missions.Calculus I is also used extensively in engineering, especially in the design and analysis of systems. For example, in electrical engineering, calculus is used to determine the power consumed by a circuit, while in civil engineering, it is used to calculate the stress on structures such as bridges and buildings. Calculus is also essential in chemical engineering, where it is used to determine therate of chemical reactions.In economics, calculus is used to model and analyze various economic phenomena, such as supply and demand, consumer behavior, and production optimization. The concepts of calculus are essential in understanding the dynamics of markets and the behavior of firms in different situations.Calculus I has numerous real-life applications, from modeling the growth of populations to understanding the spread of diseases. It is used in biostatistics to determine the probability of an individual developing a certain disease and in epidemiology to model the spread of infectious diseases. In ecology, calculus is used to study predator-prey relationships and competition between species.In the field of finance, calculus is used to determine the value of financial securities such as stocks and bonds. Understanding the concepts of calculus is essential in the field of quantitative finance, which involves using mathematical models to predict the behavior of financial markets.Overall, Calculus I is a fundamental course in mathematics that teaches students the basic concepts of differential calculus, including limits and derivatives, and their applications in various fields. It provides a solid foundation for more advanced courses in Calculus and other related fields. The concepts covered in Calculus I have numerous applications in many fields, including physics, engineering, economics, and biology, making it an essential tool for solving real-world problems.。
微积分介值定理的英文
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.。
situation calculus, discrete time
Multi-agent actions under uncertainty: situation calculus,discrete time,plans andpoliciesDavid PooleDepartment of Computer ScienceUniversity of British ColumbiaV ancouver,B.C.,Canada V6T1Z4poole@cs.ubc.cahttp://www.cs.ubc.ca/spider/pooleApril23,1997AbstractWe are working on a logic to combine the advantages offirst-order logic, but using Bayesian decision theory(or more generally game theory)as a ba-sis for handing uncertainty.This forms a logic for multiple agents under un-certainty.These agents act asynchronously,can have their own goals,have noisy sensors,and imperfect effectors.Recently we have developed the in-dependent choice logic that incorporates all of these features.In this paper we discuss two different representations of time within this framework:the situation calculus and what is essentially the event calculus.We show how they both can be used,and compare the different ontological commitments made by each.Uncertainty is handled in terms of a logic which allows for independent choices and a logic program that gives the consequences of the choices.There are probabilities over the choices by nature.As part of theconsequences are a specification of the utility of(final)states.In the situa-tion calculus,agents adopt programs and programs lead to situations in pos-sible worlds(which correspond to the possible outcomes of complete histo-ries);given a probability distribution over possible worlds,we can get theexpected utility of a program.In the event calculus view,actions are propo-sitions and agents adopt policies which are logic programs to imply what theagent will do based on what it observes.Again the expected value of a policycan be computed.The aim is to choose the plan or policy that maximizes theexpected utility.This paper overviews both approaches,and explains why Ithink the event calculus is the most promising approach.1Introduction1.1Logic and UncertaintyThere are many normative arguments for the use of logic in AI(see e.g.,Nilsson 1991,Poole,Mackworth&Goebel1997).These arguments are usually based on reasoning with symbols with an explicit denotation,allowing relations amongst individuals,and quantification over individuals.This is often translated as need-ing(at least)thefirst-order predicate calculus.However,thefirst-order predicate calculus has very primitive mechanisms for handling uncertainty.There are also normative arguments for Bayesian decision theory and game theory(see e.g.,V on Neumann&Morgenstern1953,Savage1972)for handling uncertainty.These are based on theorems that show that,under certain reasonable assumptions,rational decision makers will act as though they are using probabili-ties and utilities and maximizing their expected utilities.Game theory(Von Neu-mann&Morgenstern1953,Myerson1991,Fudenberg&Tirole1992)is the ex-tension of Bayesian decision theory to include multiple intelligent agents.We would like to combine the advantages of logic with those of Bayesian de-cision/game theory.The independent choice logic(ICL)(Poole1997)is designed with the goal of including the advantages of logic,but handling all uncertainty us-ing Bayesian decision or game theory.The idea is to not use disjunction,but rather to allow agents,including nature, to make choices from a choice space,and use a restricted underlying logic to spec-ify the consequences of the choices.To start off without disjunction,we can adopt acyclic logic programs(Apt&Bezem1991)as the underlying logical formalism. What is interesting is that simple logic programming solutions to the frame prob-lem(see e.g.,Shanahan1997,Chapter12)seem to be directly transferrable to the2ICL which has more sophisticated mechanisms for handling uncertainty than the predicate calculus.I would even dare to venture that the main problems with for-malizing action within the predicate calculus arise because of the inadequacies of disjunction to represent the sort of uncertainty we need.When mixing logic and probability,one can extend a rich logic with probabil-ity,and have two sorts of uncertainty:that uncertainty from the probabilities and that from disjunction in the logic(Bacchus1990,Halpern&Tuttle1993).An al-ternative that is pursued in the independent choice logic is to have all of the uncer-tainty in terms of probabilities.The underlying logic is restricted so that there is no uncertainty in the logic1;every set of sentences has a unique model.In particular we choose the logic of acyclic logic programs under the stable model semantics; this is a practical language with the unique model property.All uncertainty is han-dled by what can be seen as independent stochastic mechanisms.A deterministic logic program that gives the consequences of the agent’s choices and the random outcomes.In this manner we can get a simple mix of logic and Bayesian decision theory(Poole1997).1.2Actions and UncertaintyThe combination of decision or game theory and planning(Feldman&Sproull 1975)is very appealing.The general idea of planning is to construct a sequence of steps,perhaps conditional on observations that solves a goal.In decision-theoretic planning,this is generalised to the case where there is uncertainty about the envi-ronment and we are concerned,not only with solving a goal,but what happens under any of the contingencies.Goal solving is extended to the problem of max-imizing the agent’s expected utility,where the utility is an arbitrary function of thefinal state(or the accumulation of rewards received earlier).Moreover in the multi-agent case,each agent may have a different utility,they carry out actions asynchronously,and the actions of all of the agents affect the utility for each agent. Moreover there is uncertainty about the environment,and it is often optimal for an agent to act stochastically.It is our goal to write a logic that allows for prac-tical reasoning for domains that include multiple agents,utilities,uncertainty and stochastic actions.Within the independent choice logic,we have considered both the situation cal-culus(Poole1996),discrete time(Poole1997),and continuous time(Poole1995). In this paper we compare the situation calculus(McCarthy&Hayes1969)and the discrete time frameworks in this framework.Recently there have been claims made that Markov decision processes(MDPs) (Puterman1990)are the appropriate framework for developing decision theoretic planners(e.g.,Boutilier,Dearden&Goldszmidt1995).MDPs,and dynamical sys-tem in general(Luenberger1979)are based on the notion of a state:what is true at a time such that the past at that time can only affect the future from that time by affecting the state.In terms of probability:the future is independent of the past given the state.This is called the Markov property.In the most general frame-work the agents individually have states(these are called belief states)and idea is to construct a state transition function that specify how the agent’s belief state is updated from its previous belief state and its observations,and a command func-tion(policy)that specifies what the agent should do based on its observations and belief state(Poole et al.1997,Chapter12).In fully observable MDPs,the agent can observe the actual state and so doesn’t need belief states.In partially observ-able MDPs,the belief state is a probability distribution over the actual states of the system,and the state transition function is given by Bayes’rule.In between these are agents that have limited memory or limited reasoning capabilities.For a representation for time,actions and states we can adopt a number of dif-ferent representations.One is to consider robot programs as policies(Poole1995) where actions are propositions,and agents adopt logic programs that specify what they will do based on their observations and belief state.An alternative is to rep-resent actions in the situation calculus,where the agents have conditional plans (Poole1996).Both are discussed here.All of the uncertainty in our rules is relegated to independent choices as in the independent choice logic(Poole1997)(an extension of probabilistic Horn abduc-tion(Poole1993)to include multiple agents and negation as failure).2The Independent Choice LogicAn independent choice space theory is make of two principal components:a choice space that specifies what choice can be made,and a set of facts that specifies what follows from the choices.Definition2.1A choice space is a set of sets of ground atomic formulae,such that if,and are in the choice space,and then.An element4of a choice space is called a choice alternative(or sometimes just an alternative). An element of a choice alternative is called an atomic choice.Definition2.2Given choice space,a selector function is a mapping such that for all.The range of selector function,written is the set.The range of a selector function is called a to-tal choice.In other words,a total choice is a selection of one member from each element of.Definition2.3The Facts,,are sentences in some base logic with the following two properties:For every selector function,the set of sentences is definitive on every proposition;that is,for every proposition either or,where is logical consequence in the underlying logic.For all atomic choices,iff.The restriction really means that the base logic contains no uncertainty.All uncer-tainty is handled in the choice space.The semantics of an ICL is defined in terms of possible worlds.There is a pos-sible world for each selection of one element from each alternative.The atoms which follow using the consequence relation from these atoms together with are true in this possible world.Definition2.4Suppose we are given a base logic and ICL theory.For each selector function there is a possible world.We write,read“is true in world based on”,iff.When understood from context,the is omitted as a subscript of.The fact that every proposition is either true or false in a possible world follows from the definitiveness of the base logic.Note that,for each alternative and for each world,there is exactly one element of that’s true in.In particular,,and for all .The base logic we use is that of acyclic logic programs(Apt&Bezem1991), such that no atomic choice is in the head of any rule.means is true in the (unique)stable model of.This logic is definitive in the above sense and is rich enough to axiomatise many of the domains we are interested in.Note that acyclic logic programs allow recursion,but all recursion must be well founded.This semantic construction is the core of the ICL.Other components we require for different applications include:5is afinite set of agents.There is a distinguished agent called“nature”.is a function from.If then agent is said to control alternative.If is an agent,the set of alternatives controlled by is.Note that.is a function such that,.That is,for each alternative controlled by nature,is a probability measure over the atomic choices in the alternative.When each agent(other than nature)makes a choice(possibly stochastic)from each alternative it controls,we can determine the probability of any proposition. The probability of a proposition is defined in the standard way.For afinite choice space2,the probability of any proposition is the sum of the probabilities of the worlds in which it is true.The probability of a possible world is the product of the probabilities of the atomic choices that are true in the world.That is,the atomic choices are(unconditionally)probabilistically independent.Poole(1993)proves that such independent choices together with an acyclic logic program can represent anyfinite probability distribution.Moreover the structure of the rule-base mirrors the structure of Bayesian networks(Pearl1988)3.Similarly we can define the ex-pectation of a function that has a value in each world,as the value averaged over all possible worlds,weighted by their probability.See Poole(1997)for more details on the ICL.3The Situation Calculus and the ICLIn this section we sketch how the situation calculus can be embedded in the ICL. What we must remember is that we only need to axiomatise the deterministic as-pects in the logic programs;the uncertainty is handled separately.What gives us confidence that we can use simple solutions to the frame problem,for example,is that every statement that is a consequence of the facts that doesn’t depend on the atomic choices is true in every possible world.Thus,if we have a property thatdepends only on the facts and is robust to the addition of atomic choices,then it will follow in the ICL;we would hope than any logic programming solution to the frame problem would have this property.Before we show how to add the situation calculus to the ICL,there are some design choices that need to be made even to consider just single agents.In the deterministic case,the trajectory of actions by the(single)agent up to some time point determines what is true at that point.Thus,the trajectory of actions,as encapsulated by the‘situation’term of the situation calculus (McCarthy&Hayes1969,Reiter1991)can be used to denote the state,as is done in the traditional situation calculus.However,when dealing with un-certainty,the trajectory of an agent’s actions up to a point,does not uniquely determine what is true at that point.What random occurrences or exoge-nous events occurred also determines what is true.We have a choice:we can keep the semantic conception of a situation(as a state)and make the syntactic characterization more complicated by perhaps interleaving exoge-nous actions,or we can keep the simple syntactic form of the situation calcu-lus,and use a different notion that prescribes truth values.We have chosen the latter,and distinguish the‘situation’denoted by the trajectory of actions, from the‘state’that specifies what is true in the situation.In general there will be a probability distribution over states resulting from a set of actions by the agent.It is this distribution over states,and their corresponding utility, that we seek to model.This division means that agent’s actions are treated very differently from ex-ogenous actions that can also change what is true.The situation terms define only the agent’s actions in reaching that point in time.The situation calculus terms indicate only the trajectory,in terms of steps,of the agent and essen-tially just serve to delimit time points at which we want to be able to say what holds.This is discussed further in Section5.None of our representations assume that actions have preconditions;all ac-tions can be attempted at any time.The effect of the actions can depend on what else is true in the world.This is important because the agent may not know whether the preconditions of an action hold,but,for example,may be sure enough to want to try the action.When building conditional plans,we have to consider what we can condi-tion these plans on.We assume that the agent has passive sensors,and that7it can condition its actions on the output of these sensors.We only have one sort of action,and these actions only affect‘the world’(which includes both the robot and the environment).All we need to do is to specify how the agent’s sensors depend on the world.This does not mean that we can-not model information-producing actions(e.g.,looking in a particular place)—these information producing actions produce effects that make the sensor values correlate with what is true in the world.The sensors can be noisy;the value they return does not necessarily correspond with what is true in the world(of course if there was no correlation with what is true in the world, they would not be very useful sensors).Before we introduce the probabilistic framework we present the situation cal-culus(McCarthy&Hayes1969).The general idea is that robot actions take the world from one‘situation’to another situation.We assume there is a situation that is the initial situation,and a function4that given action and a situa-tion returns the resulting situation.An agent that knows what it has done,knowswhat situation it is in.It however does not necessarily know what is true in that sit-uation.The robot may be uncertain about what is true in the initial situation,what the effects of its actions are and what exogenous events occurred.3.1The ICL SCWithin the ICL we can use the situation calculus as a representation for change. Within the logic,there is only one agent,nature,who controls all of the alterna-tives.These alternatives thus have probability distributions over them.The prob-abilities are used to represent our ignorance of the initial state and the outcomes of actions.We can then use the situations to reflect the“time”at which somefluents are true or not.We model all randomness as independent stochastic mechanisms,such that an external viewer that knew the initial state(i.e.,what is true in the situation),and knew how the stochastic mechanisms resolved themselves would be able to pre-dict what was true in any situation.Given a probability distribution over the initial state and the stochastic mechanisms,we have a probability distribution over the effects of actions.This is modelled by having the mechanisms as atomic choices controlled by nature(and so with a probability distribution).We use logic to specify the transitions specified by actions and thus what is true in a situation.What is true in a situation depends on the action attempted,what was true before and what stochastic mechanism occurred.Afluent is a predicate (or function)whose value in a world depends on the situation;we use the situation as the last argument to the predicate(function).We assume that for eachfluent we can axiomatise in what situations it is true based on the action that was performed, what was true in the previous state and the outcome of the stochastic mechanism.Note that a possible world in this framework corresponds to a complete history.A possible world specifies what is true in each situation.In other words,given a possible world and a situation,we can determine what is true in that situation. Example3.1We can write rules such as,the robot is carrying the key after it has (successfully)picked it up:is true if the agent would succeed if it picks up the key and is false if the agent would fail to pick up the key.The agent typically does not know the value ofwould be an atomic choice.That is5This is now a reasonably standard logic programming solution to the frame problem(Shanahan 1997,Chapter12),(Apt&Bezem1991).It is essentially the same as Reiter’s(1991)solution to the frame problem.It is closely related to Kowalski’s(1979)axiomatization of action,but for each proposition,we specify which actions are exceptional,whereas Kowalski specifies for every every action which propositions are exceptional.Kowalski’s representation could also be used here.9Example3.2For example,an agent is carrying the key as long as the action was not to put down the key or pick up the key,and the agent did not accidentally drop the key while carrying out another action:may be something that the agent does not know whether it is true—there may be a probability that the agent will drop the key.If dropping the key is independent at each situation,we can model this as:The prize depends on whether the robot reached its destination or it crashed.No matter what the definition of any other predicates is,the following definition of will ensure there is a unique prize for each world and situation:The resources used depends not only on thefinal state but on the route taken.To model this we make afluent,and like any otherfluent we axiomatise it: Here we have assumed that non-goto actions cost,and that paths have costs. Paths and their costs can be axiomatised usingthat is true if the path from to via has cost.3.3Axiomatising SensorsWe also need to axiomatise how sensors work.We assume that sensors are pas-sive;this means that they receive information from the environment,rather than “doing”anything;there are no sensing actions.This seems to be a better model of actual sensors,such as eyes or ears,and makes modelling simpler than when sensing is an action.So called“information producing actions”(such as opening the eyes,or performing a biopsy on a patient,or exploding a parcel to see if it is (was)a bomb)are normal actions that are designed to change the world so that the sensors correlate with the value of interest.Note that under this view,there are no information producing actions,or even informational effects of actions;rather var-ious conditions in the world,some of which are under the robot’s control and some of which are not,work together to give varying values for the output of sensors.11A robot cannot condition its action on what is true in the world;it can only condition its actions on what it senses and what it remembers.The only use for sensors is that the output of a sensor depends,perhaps stochastically,on what istrue in the world,and thus can be used as evidence for what is true in the world.Within our situation calculus framework,can write axioms to specify how sensed values depend on what is true in the world.What is sensed depends on the situa-tion and the possible world.We assume that there is a predicate that is true if is sensed in situation.Here is a term in our language,that representsone value for the output of a sensor.is said to be observable.Example3.4A sensor may be to be able to detect whether the robot is at the sameposition as the key.It is not reliable;sometimes it says the robot is at the same po-sition as the key when it is not(a false positive),and sometimes it says that the robot is not at the same position when it is(a false negative).Suppose that noisy sensoris true(in a world)if the robot senses that it is at the key in situation.It can be axiomatised as:Thefluent is true if the sensor is giving a false-positive value in situation,and is true if the sensor is not giving a false negative in situation.Each of these could be part of an atomic choice,which would let us model sensors whose errors at different times are independent.If we had a theory about how sensors break,we could write rules that imply these fluents.123.4Conditional PlansThe idea behind the ICL SC is that agents get to choose situations(they get to choose what they do,and when they stop),and‘nature’gets to choose worlds(there is a probability distribution over the worlds that specifies the distribution of effects ofthe actions).Agents do not directly adopt situations,they adopt‘plans’or‘programs’.In general these programs can involve atomic actions,conditioning on observations, loops,nondeterministic choice and procedural abstraction(Levesque,Reiter,Lesp´e rance, Lin&Scherl1996).In this paper we only consider simple conditional plans whichare programs consisting only of sequential composition and conditioning on obser-vations(Levesque1996,Poole1996)).An example conditional plan is:if then else endIfAn agent executing this plan will start in situation,then do action,then it will sense whether is true in the resulting situation.If is true,it will do then,and if is false it will do then then.Thus this plan either selects the situa-tion or the situation.It selectsthe former in all worlds where is true,and selects the latter inall worlds where is false.Note that each world is definitive on eachfluent for each situation.The expected utility of this plan is the weighted av-erage of the utility for each of the worlds and the situation chosen for that world. The only property we need of is that its value in situation will be able tobe observed.The agent does not need to be able to determine its value beforehand.Definition3.5A conditional plan,or just a plan,is of the formwhere is a primitive actionwhere and are plansif then else endIfwhere is observable;and are plansNote that“”is not an action;the plan means that the agent does not do anything—time does not pass.This is introduced so that the agent can stop with-out doing anything(this may be a reasonable plan),and so we do not need an“if then endIf”form as well;this would be an abbreviation for“if then elseendIf”.13Plans select situations in worlds.We can define a relation:that is true if doing plan in world from situation results in situation. This is similar to the macro of Levesque et al.(1996)and the of Levesque (1996),but here what the agent does depends on what it observes,and what the agent observes depends on which world it happens to be in.We can define the relation in pseudo Prolog as:if then else endIfif then else endIfNow we are at the stage where we can define the expected utility of a plan.The expected utility of a plan is the weighted average,over the set of possible worlds, of the utility the agent receives in the situation it ends up in for that possible world: Definition3.6If our theory is utility complete,the expected utility of plan is6:(summing over all selector functions on)whereifwhere(this is well defined as the theory is utility complete),andis the utility of plan P in world.is the probability of world. The probability is the product of the independent choices of nature.It is easy to show that this induces a probability measure(the sum of the probabilities of the worlds is).4Independent Choice Logic and Reactive PoliciesThere is a completely different way to use the ICL to model time and action.Here we can only sketch the idea;see Poole(1997)for details.We only consider discrete time here.The idea is to made agents and nature in the same way.For the situation cal-culus axiomatization above,the single agent was treated in a completely different way to nature.Symmetry is important when we consider multiple agents.We represent time in terms of the integers.The fact that the agent attempted an an action is represented a propositions indexed by time.We can use a predicate that is true if the agent attempted action at time.What is true at a time depends on what was true at the previous times and what actions have occurred, and the outcome of stochastic mechanisms.This places actions by the agent at the same level as actions by nature.There are two parts to axiomatise.Thefirst is to axiomatise the effect of ac-tions,and the second is to specify what an agent will do based on what it observes (i.e.,its policy).To axiomatise the effect of actions,for the discrete time case we can simply write how what is true at one time depends on what was true at the previous time (including what actions occurred).We would write similar axioms to the situation calculus,but indexed by time,and using as a predicate.Example4.1The axiom for carrying of Example3.1can be stated as:The frame axiom for in Example3.2would look like:Shanahan(1997)in that it is reasoning about a particular course of events.This is true for each possible world,but we can have a probability distribution over possible worlds.We have a mechanism for allowing multiple agents to choose what events that they can control occur,and to allow a probability distribution over events that nature controls.For each world,we only need to worry about what events are true in that world.5DiscussionWe have given a(too)brief sketch of two different representations of change for a single agent under uncertainty in the ICL.See Poole(1997)and Poole(1996)for more details.In some sense the axioms look similar;there is not really much difference be-tween the situation calculus and event calculus axiomatization given here.The ma-jor difference is that the robot programs that are natural for the situation calculus are very different from the reactive policies that are natural for the event calculus.When we extend this to multiple agents,and stochastic actions(as is often needed for agents with limited sensing and communication),the event calculus framework can easily be adapted.Nothing needs to be changed to allow for concurrent actions (the actions by each agent).Each agent adopts its own(private)policy based on what it can sense and what it can do.Extending the situation calculus version to multiple agents isn’t so straightfor-ward.The way we have treated the situation calculus(and we have tried hard to keep it as close to the original as possible)really gives an agent-oriented view of time—the‘situations’in some sense mark particular time points that correspond to the agent completing its actions.Everything else(e.g.,actions by nature or other agents)then has to meld in with this division of time.This is even trickier when we realize that when agents have sloppy actuators and noisy sensors,the actions defining the situations correspond to action attempts;the agent doesn’t really know what it did it only knows what it attempted and what its sensors now tell it.When there are multiple agents,either there has to be a common clock,we need some master agent which the other agents define their state transition,or complex ac-tions(Reiter1996,Lin&Shoham1995).These all mean that the actions need to be carried out lock-step,removing the intuitive appeal of the situation calculus,and making it much closer to the event calculus.The work of Reiter(1996)and Lin& Shoham(1995)assumes a very deterministic world.Not only must the world un-fold deterministically,but you must know how it unfolds.This is very different17。
precalculus知识点总结
precalculus知识点总结Precalculus is an essential branch of mathematics that serves as a bridge between algebra, geometry, and calculus. This subject is crucial for students preparing to undertake advanced courses in mathematics, physics, engineering, and other technical fields. In this precalculus knowledge summary, we will cover important topics such as functions, trigonometry, and analytic geometry.FunctionsOne of the fundamental concepts in precalculus is that of functions. A function is a relationship between two sets of numbers, where each input is associated with exactly one output. In other words, it assigns a unique value to each input. Functions can be represented in various forms, such as algebraic expressions, tables, graphs, and verbal descriptions.The most common types of functions encountered in precalculus include linear, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric functions. Each type of function has its own unique characteristics and properties. For example, linear functions have a constant rate of change, while quadratic functions have a parabolic shape.Functions can be manipulated by performing operations such as addition, subtraction, multiplication, division, composition, and inversion. These operations can be used to create new functions from existing ones, or to analyze the behavior of functions under different conditions.TrigonometryTrigonometry is the study of the relationships between the angles and sides of triangles. It plays a crucial role in precalculus and is essential for understanding periodic phenomena such as oscillations, waves, and circular motion.The primary trigonometric functions are sine, cosine, and tangent, which are defined in terms of the sides of a right-angled triangle. These functions have various properties, such as periodicity, amplitude, and phase shift, which are important for modeling and analyzing periodic phenomena.Trigonometric functions can also be extended to the entire real line using their geometric definitions. They exhibit various symmetries and periodic behaviors, which can be visualized using the unit circle or trigonometric graphs. Additionally, trigonometric identities and equations are essential tools for simplifying expressions, solving equations, and proving theorems.Analytic GeometryAnalytic geometry is a branch of mathematics that combines algebra and geometry. It deals with the use of algebraic techniques to study geometric shapes and their properties. Inprecalculus, this subject is primarily concerned with the study of conic sections, such as circles, ellipses, parabolas, and hyperbolas.The equations of conic sections can be derived using geometric constructions, or by using algebraic methods such as completing the square, factoring, and manipulating equations. These equations can then be used to describe the geometric properties of conic sections, such as their shape, size, orientation, and position.Furthermore, analytic geometry also involves the study of vectors and matrices, which are important tools for representing and manipulating geometric objects in higher dimensions. Vectors can be used to represent points, lines, and planes in space, while matrices can be used to perform transformations such as rotations, reflections, and scaling.Other TopicsIn addition to the core topics mentioned above, precalculus also covers other important concepts such as complex numbers, polar coordinates, sequences and series, and mathematical induction. Complex numbers are used to extend the real number system to include solutions to equations that have no real roots. They have applications in various fields such as electrical engineering, quantum mechanics, and signal processing.Polar coordinates provide an alternative way of describing points in the plane using radial distance and angular direction. They are particularly useful for representing periodic and circular motion, as well as for simplifying certain types of calculations in calculus.Sequences and series are ordered lists of numbers that have a specific pattern or rule. They can be finite or infinite, and their sums can be used to represent various types of mathematical and physical phenomena. For example, arithmetic sequences are used to model linear growth or decline, while geometric series are used to model exponential growth or decay.Finally, mathematical induction is a powerful method for proving statements about positive integers. It is based on the principle that if a certain property holds for a base case, and if it can be shown that it also holds for the next case, then it holds for all subsequent cases as well. This method is widely used in various areas of mathematics, such as number theory, combinatorics, and discrete mathematics.ConclusionIn conclusion, precalculus is a diverse and rich subject that covers a wide range of mathematical concepts and techniques. It provides students with the necessary foundation to tackle more advanced topics in calculus and beyond. By mastering the core topics of precalculus, students will be well-equipped to understand and apply advanced mathematical methods in various technical fields. Whether it be functions, trigonometry, analytic geometry, or any other topic, a solid understanding of precalculus is essential for success in higher mathematics.。
量子计算机 经典问题
量子计算机经典问题【中英文版】Title: Quantum Computers and Classical QuestionsTitle: 量子计算机与经典问题The phrase "quantum computer" often evokes images of futuristic technology, capable of solving complex problems beyond the scope of classical computers.However, the concept of quantum computing also raises classical questions in the field of computer science.“量子计算机”这一词汇常引发人们对未来科技的想象，能够解决传统计算机难以应对的复杂问题。
然而，量子计算的概念在计算机科学领域也引发了经典问题。
One of the classical questions in computer science is the "halting problem," which asks whether it is possible to determine, for an arbitrary program, whether it will eventually halt.This problem is known to be undecidable in the classical computing paradigm.However, quantum computing offers a potential solution to this problem.计算机科学中的一个经典问题是“停机问题”，它询问是否可以确定任意程序最终是否会停止。
这个问题在传统计算范式中已知是不可判定的。
《微积分英文版》课件
Limits and continuity
Definition: A limit is the value that a function approaches as the input approaches a certain point Continuity means that the function doesn't have any breaks or jumps at any point
Course structure
03
The course is divided into several modules, each focusing on a specific topic in calculus Learners can complete the course at their own pace and in any order of the modules
Properties: One side limits, absolute continuity, uniform continuity, etc
Differentiation
Definition: The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point It can be used to find the rate of change of a function
Integral definition: The integral of a function is a measure of the area under its curve It is calculated by finding the limit of the sum of areas of rectangles under the curve as the width of the rectangles approaches zero
IOS Press Logical Analysis of Biological Systems ￡
Fundamenta Informaticae64(2005)271–285271 IOS PressLogical Analysis of Biological SystemsRadu Mardare and Corrado PriamiDepartment of Information and Communication TechnologyTrento University,Italymardare@dit.unitn.itAbstract.Our paper proposes a technique for performing logical analysis over the calculi for com-munication and mobility,i.e.,Ambient Calculus type of calculi.We show how this analysis can beused in the case of biological models in order to obtain significant information for biologists.The technique is based on set theoretical models we developed for ambient processes by using thepower of Hypersets Theory.These models are further used as possible worlds in a Kripke structureorganized for a propositional branching temporal logic.Providing the temporal logical structure for the accessibility relation between ambient processes,we open the perspective of reusing model checking algorithms developed for temporal logics inanalyzing any phenomena that can be described by these calculi.Keywords:Hypersets,process algebra,ambient calculus,temporal logics,model checking1.IntroductionAmbient Calculus[11]is a useful tool to construct mathematical models for complex systems because of its facilities in expressing hierarchies of locations and their mobility.It was developedfirst as a natural extension,with locations,of-Calculus[22,24],in order to provide models for phenomena concerning communication and ter,it was adapted to model biological systems[28,9],as an algebraical alternative to the Membrane Computing approach[25].Work partially supported by the FET project IST-2001-32072DEGAS under the pro-active initiative on Global Computing. Address for correspondence:Via Sommarive14,I-38050POVO,Trento(TN)–ItalyAlso work:Bucharest University,Romania272R.Mardare and C.Priami/Logical Analysis of Biological SystemsEven if such a calculus is able to simulate the behavior of some biological systems,for giving to biologists a real useful tool,we need to do more.It could be useful to develop some techniques for making predictions over the systems,such that the results of some expensive biological experiments to be predicted by some computable formal analysis of the mathematical models.Properties such as the protein will split,or there is a possible future where the complexAB precedes the proteinA are not expressible inside Ambient Calculus.Only a logic built on top of it can describe such phenomena.In order to answer such requirements Ambient Logic[12,13]and Spatial Logics[10]were devel-oped.These logics can describe properties of mobile computations as well as the hierarchy of locations and modifications of this hierarchy in time.The main idea of these is to treat processes as spatio-temporal entities,thus two types of modalities have been used–one for assertions about space and the other for assertions about time.This paper,resuming our results presented in[19,20],proposes a propositional branching temporal logic,,constructed on top of Ambient Calculus,as a more convenient alternative to Spatial Logics. Temporal logics have emerged,lately,in many domains as a good compromise between expressiveness and abstraction.Many of them support useful computational applications as model checking.For the particular cases of or,these techniques were developed up to the construction of some tools able to perform such analysis(see,e.g.,SMV[4],NuSMV[2],HyTech[1],VIS[5]).The main feature of our logic is that thefinal state of any computation can be reconstructed by just having information about the initial state and the history of the computation.The spatial structure of a state is fully described by a set of atomical propositions,while the possible states are described using,in addition,a temporal modality.In this respect our approach is different from those used in Ambient Logic,or Spatial Logics, giving us the advantages of simplicity and expressivity that a CTL*logic has with respect to the cited modal logics.Moreover,seems unproblematic to extend the same sort of logic for other calculi in this paradigm,e.g.,BioAmbients Calculus[28],or Brane Calculi[9].Being the peculiarities of the calculi of communication,it was more profitable for us to develop the logic not directly for Ambient Calculus,but for a set theoretical model of it.This model goes further with Peter Aczel’s idea of developing a model for Milner’s calculus of communication systems CCS [21].Aczel’s try wasfinalized with the development of Hypersets Theory[6],and we found it useful to use these hypersets to model communication systems with locations,as Ambient Calculus.The rest of the paper is organized as follows.We introducefirst the Ambient Calculus and we discuss some special features of it.In the third section we present a couple of simple case studies coming from biology.They are used to comment on the advantages of applying temporal logics to the Ambient Calculus specification of phenomena related to life sciences.Section four introduces the theoretical underpinning of our logic:labelled syntax trees which are the graph presentation of the set theoretical model for the ambients.In Section5we define a branching temporal logic for Ambient Calculus,and show how to run simple reachability properties on our case studies.Thefinal section concludes the presentation with some general remarks and a sketch of the future research directions we intend to take.2.Overview on Ambient CalculusFirst,we briefly recall the Ambient Calculus[11]starting with the syntax of ambient processes.R.Mardare and C.Priami/Logical Analysis of Biological Systems273::=processes::=capabilitiesrestriction namevoid can enter into Mcomposition can exit out of Mreplication can open Mambient pathcapability action nullinput actionoutput actionWe accept,in addition to the previous definition,that ambient programs can include unspecified processes denoted by capital letters P,Q and R,hereafter named atomical processes1.Let be the class of atomical processes and the class of ambients.A structural congruence is defined over processes by:ifIn addition,we identify processes up to renaming of bound names:ifFinally,the operational semantics of the ambient calculus is defined by the rules:1This is a necessary requirement in developing complex analysis,as model checking,for Ambient Calculus because we have to recognize and distinguish over time,unspecified processes inside the target process.For instance,is an unspecified process in274R.Mardare and C.Priami/Logical Analysis of Biological Systems2.1.Handling the new namesConsider the next process:Here,means that the name inside the scope of is different from all the other names in the program.In the example,we want to be sure that and,which are capabilities prefixing the process,will never act over but only over the ambient that was chosen to name thefirewall.Vice versa,,the capability of,will never act over thefirewall ambient,but only over.The intuition is that the name of the ambient chosen to name thefirewall should be one that has not been used before.A possible solution could be just to choose a new name and to replace with it in all its occurrences inside the scope of.This solution is locally good,but it will not prevent the name from ever being used in other processes that we could combine with ours,so that a name conflict would arise.In other words,the renaming solution is not a compositional one.We guarantee compositionality by a trick that resembles de Bruijn indexes for name-free-calculus:we accept ordered pairs of natural numbers as possible names of ambients and we use them to completely remove any occurrence from processes.So,we replace the new name in a process with2the pair.This approach allows us to combine our process with others for which we already constructed the labelled syntax trees. In this way all the names in the second process will receive names as meaning that is the new name of the second process,and so on,the new name of the process will receive the name.This construction is supported by the assumption that inside an ambient process can only occur a finite number of new name operators and that we will combine only afinite number of processes.According with the above,our example becomes:The analysis of the reductions of our process shows that the expected result is still possible without using the new name operator.Indeed:Hereafter we will treat as being an ambient name,whenever it appears in our processes.This means that the set contains,as a subset,a subset of.This modification does not affect the four rules of structural congruence((Struct Res Res),(Struct Res Par),(Struct Res Amb)and(Struct Zero Res),see[11]).Only it modifies the intentional interpretation of.It will not mean this name is new inside the scope of our quantifier,but replace this name in all its occurrences inside the scope of our quantifier by an unused pair of natural numbers.In this way we reduce all the syntax trees of ambient calculus to syntax trees without new name operators.2We will replace in the ambient calculus process,all the occurrences of inside the scope of,being ambients or capabili-ties,withR.Mardare and C.Priami/Logical Analysis of Biological Systems275 ing Ambient Calculus to model biological phenomenaIn the cited literature,many examples of biological phenomena described using Ambient Calculus type of calculi can be found.In this section we will present two such examples,described in[20],that we found relevant for the advantages of a logical analysis.In thefirst example we will describe a part of the scene of the interaction between a Virus and a Macrophage.The Macrophage,by its structure,is able to recognize a Virus by its characteristics.Once it is recognized,the Virus is engulfed by Macrophage and is destroyed.Wefind it appropriate to describe the Macrophage as an ambient named that contains a process able to destroy the virus;the virus is an ambient that contains inside a process.The Macrophage recognizes the virus by the name and by its structure(i.e.,Macrophage knows the names that define the structure of the virus).Using this information,Macrophage manages to put in parallel the processes andand in this way annihilates the action of the virus.We can describe this action in Ambient Calculus in a way similar with the description of the action of afirewall[19]:For this situation,we are interested in the success of our system,in all possible time paths,to achieve the state where the processes and are in parallel inside the ambient(that represent the Macrophage),such that the virus is annihilated.If our system succeeds to do this,we can say that is an appropriate model for the biological phenomenon,otherwise we have to reconsider our approach.Such properties,we will argue further,can be naturally expressed using a temporal logic.Consider now the second example,the model of the trimetric GTP binding proteins(G-proteins)that plays an important role in the signal transduction pathway for numerous hormones and neurotransmitters [3,7].It consists offive processes:a regulatory molecule,a receptor,and three domains that are bound together composing the protein and.Data sent by to determine a communication between the receptor and the protein that causes the breakage of the boundary of and.We can express this in Ambient Calculus by the following specification:,,,where is a name that appear in only,bounded by the input prefix276R.Mardare and C.Priami/Logical Analysis of Biological Systemswhere we denoted the process obtained by substituting with inside by.In this example we accepted recursive definitions in Ambient Calculus.This pushed us outside the classical syntax of this calculus.Still the model and the logic we will introduce further can handle with such an extension.Anyway,if we really want to describe the model strictly in the classical syntax,we could differentiate the recursion variables by using some indexes.With respect to the above example we are interested in expressing properties like for all possible future paths,sometime in the future,we will have the interaction that will generate the split of the protein.One might also want to express that the protein will not be split before the interaction between and will be performed(a property will not be satisfied until another one will be).Both properties above are examples of‘temporal’properties.4.The labelled syntax treesIn order to define the temporal logic,we reorganize the spatio-temporal information contained by an ambient process.This will be done by defining a special labelling function for the syntax trees of Ambient Calculus.A syntax tree for a process is a graph withwhereis a set that contain all the unspecified process nodes(the atomical processes collected in the subset )and the ambient nodes(collected in the subset);is the set of capability nodes(we include here the input nodes and the nodes of variables over capabilities as well);andis the set of syntactical operator nodes(this set contains the parallel operators and the prefix operators,).We identify the subset of the prefix nodes that are immediately followed,in the syntax tree,by the parallel operator because they play an important role in the spatial structure of the ambient process3.We consider also the possibility of having circular branches in our trees,when recursive definitions are involved.All the further discussion is including these cases as well.The intuition behind the construction of a labelled syntax tree is to associate to each node of the syntax tree some labels by two functions:that gives to each node an identity,and that registers the spatial position of the node.The identity function associates a label(urelement or):1.to each unspecified process and to each ambient;this label will identify the node and will help usfurther to distinguish between processes that have the same name,2.to each capability,the identity of the process in front of which this capability is placed,3.,to each syntactical node.3These point operators are those that connect a capability with a process formed by a parallel composition of other processes bounded together by brackets,hereafter complex processes,as inR.Mardare and C.Priami/Logical Analysis of Biological Systems277The spatial function associates:1.to each ambient the set of the identities of its children4,while to unspecified processes associatesthe-label,2.to each capability,a natural number that counts the position of this capability in the chain ofcapabilities(if any)belonging to the same process,3.to each syntactical node the spatial function associates0,except for the nodes in to whichthe function will associate the set of identities of the processes connected by the main parallel operator in the compound process that this point is prefixing;for example,in the situation,.In what follows,we choose to work in Zermelo-Fraenkel system of Set Theory ZFA with the Anti-Foundation Axiom(AFA),as being a fertilefield that offers many tools for analyzing structures,as argued in[8].This approach allows us to describe the spatial structure of ambient processes as equations in set theory,each such equation being then used as atomical proposition in our logic.In this way we will not use a modality in describing the hierarchy of locations,as Spatial Logics does,but only in describing the evolution of the hierarchy in time.Hereafter,we assume a class of urelements,set-theoretical entities which are not sets(they do not have elements)but can be elements of sets.The urelements together with the empty set will generate all the sets we will work with(sometimes sets of sets).Definition4.1.A set is transitive if all the elements of a set,which is an element of,also belong to:if then.The transitive closure of,denoted by is the smallest transitive set including.The existence of could be justified as follows:.Definition4.2.The support of a set,denoted by is.The elements of are the urelements that are somehow involved in.Definition4.3.If then is a set and.is the class of all sets in which the only urelements that are somehow involved are the urelements of.Definition4.4.Let be the syntax tree associated with the ambient process.We call the structure graph associated with,the graph obtained by restricting the edge relation of the syntax tree to,i.e.,the graph defined by:for,iff (and does not exist such that).Intuitively,the structure graph of a process is obtained by restricting the edge relation of its syntax tree to.Definition4.5.A decoration of a graph is an injective function such that for all we have:4We use the terms parent and child about processes,meaning the immediate parent and immediate child in Ambient Calculus processes.278R.Mardare and C.Priami/Logical Analysis of Biological Systemsif does not exist such that,then,if such that,then for all such that.We now introduce a set of auxiliary functions that are the block definitions for and.Definition4.6.Let the next functions be defined on the subsets of nodes of the syntax tree as follows:Let be a decoration of the structure graph associated with our syntax tree.Let be an injective function such that for all.Consider,.Let defined by(is the class of natural numbers)iffiffConsider.Let defined byLet such thatiff or withiff and withLet defined for such that byiff withiff withiffSummarizing,we can define the identity function and the spatial function by:iff iff iff iff iff iffR.Mardare and C.Priami/Logical Analysis of Biological Systems279 Observe that while the range of is,the range of is(we consider here natural numbers as cardinals5so that no structure anomaly emerges as long as).We identify the sets of urelements chosen for ambients,of urelements chosen for atomical processes,and the set of sets of urelements that contain all the addresses of the elements in.We now define labelled syntax tree for a given syntax tree of an ambient process.Definition4.7.Let be the syntax tree of the ambient process.We call the labelled syntax tree of it the triplet,where is the function defined on the nodes of the syntax tree,byfor all.Remark4.1.It is obvious the central position of the function in the previous definitions.For a particular ambient process,once we defined the function,all the construction,up to the labelled syntax tree,can be done inductively on the structure of the ambient process.Because of this,our construction of the labelled syntax tree is unique up to the choice of urelements(i.e.,of and).Definition4.8.For a given labelled syntax tree we define the functions:by:ififThis function associates to each node of the structure graph the set-theoretical identity defined by the labelled syntax tree.Let be the function defined by.It associates to each ambient and compound process the set of addresses of its children.,where is the set of names of ambients of Ambient Calculus,and is the set of atomical processes.For each,is the name of the process with which is associated by6,and if.By the function each urelement(or set of urelements)used as identity will receive the name of the ambient or atomical process that it is pointing to(the sets receive the name).5Informally,we treat0as,1as,2as,3as and so on.6Informally we could say that,on,we have,but this is not exact for the reason that is an injective function while is not.Because if we have two processes named,then,for both,the value by will be,but,by,they point to different nodes in the syntax tree.280R.Mardare and C.Priami/Logical Analysis of Biological Systemsfor each,where such that and does not exist such that and .In the case that,for we cannotfind any such,we define, being the null capability.We adopt the following enrichment of the relation of equality on capability chains defined by the next rules7:–,–,–.The function associates with each of these the list of capabilities that exists in front of the process they point to.Definition4.9.Let be a labelled syntax tree of the ambient process.We will call the canonical labelled syntax tree associated with,denoted by,the restriction of the labelled syntax tree to the set,where is the null process and is the null capability.Further we will analyze only canonical labelled trees(by extension canonical processes),these being those which evolve during the ambient calculus computations,therefore they are those which really matter for our purpose.Always,we consider ambient processes enclosed in a master ambient which stays for the environment.Being the reduction rule,there is no danger in doing this.Other aspects concerning the definition of the labelled syntax tree for situations that involves the new name operator,the replication operator,or recursive processes can be found in[18].Also we introduce an algebra of labelled trees in order to analyze their composition.In[18]we proved that the function that associates to each ambient process the setis generating a sound model for Ambient Calculus.Moreover,the tuple satisfies the requirements of the definition of aflat system of equations which can describe a hyperset uniquely up to bisimulation relation,see[8].For this reason we can interpret the tupleas a labelledflat system of equations.This is the set theoretical model for communication and mobility calculi.Such a result is important from a few points of view.First we are confident in the possibility to extend this model to all Milner’s bigraphs[23]that are seen now as the only model of communication systems.Second,replacing the structure of membership relation in the definition of theflat system of equations with a relation defined by a fuzzy set structure we hope to provide a model for any calculus of communication involving stochastic information,such as[26,27].5.The LogicThe logic we construct is a branching propositional temporal logic8,.The requirements for such a construction[14]are to organize a structure,where is the initial state of our 7These rules are allowed by the syntax of Ambient Calculus together with the rules of structural congruence over processes.8We choose because is more expressive then CTL,but a CTL is possible as wellmodel,is the class of all possible states in our model,is the accessibility relation between states, ,and is a function which associates to each state a set of atomical propositions–the set of the atomical propositions true in the state(will be the class of atomical propositions and the power-set operator).We propose to use the ordered sets as states in our logic.The choice of the initial state should depend on the purpose of our analysis.If we are interested in the future of an ambient calculus process by itself,then its ordered set will be the initial state.But if will interact with another process,or will become a child of an ambient,or both like in,then,even if we have a particular interest in,the initial state should be the ordered set of.For this purpose we defined computation operations over these ordered sets to be able,starting from the sets constructed for some initial processes,to obtain the sets for other processes constructed on top of these(for more see [18]).The construction of should be done in such a way to contain all the possible future states of the initial state.For this reason we take,where is the initial state.The intuition is that no matter how the process will evolve,it is not possible to appear in it new elements than those that already exist in the initial state9.Our main idea is to define the atomic propositions such that they express the basic equations that define the spatial relations between parts of our process.So,we could define the set of atomical propo-sitions as:.In our logic we want to be just an atomical proposition and,just letters.The cardinality of iswhich depends(polynomially)on the number of atomical processes and ambients in the ambient calculus process.Further,the interpretation function is defined by:or.As it concerns the accessibility relation,following the previous intuition we could define it for two states and,constructed for the processes and,by iff(i.e., can be reached from in one step of ambient calculus reduction).5.1.SyntaxFurther,we could introduce the syntax of the CTL*logic[14].We inductively define a class of state formulae(which will be true or false of states)and a class of path formulae10(true or false on paths), starting from.We accept as basic operators the logical operators and,the temporal operators (next time)and(until)and the path quantifier(for some futures).We will derive from them all the 9We include here also the situations where some ambients were dissolved by consuming,for example,open capability;we consider,in this case,that these ambients still exist in our process but they have an“empty position”.10A fullpath is an infinite sequence of states such that for all.We use the convention that if denotes a fullpath,then denotes the suffix path.usual propositional logic operators,the temporal operators(always)and(sometimes)and the path quantifier(for all futures).Syntactical rules:1.Each atomical proposition is a state formula.2.If are state formulae then so are,.3.If is a path formula then,are state formulae.1’.Each state formula is a path formula.2’.If are path formulae then so are,.3’.If are path formulae then so are,.Syntactical conventions:1.abbreviates.2.abbreviates.3.abbreviates.4.abbreviates.5.abbreviates.5.2.SemanticsNow we define inductively.We write to mean that the state formula is true at statein the model,and to mean that the path formula is true for the fullpath in the structure .The rules are:iff whereiff andiff it is not the case thatiff fullpath in withiff fullpath in withiffiff andiff it is not the case thatiff and impliesiffDefinition5.1.A state formula(resp.path formula)is valid provided that for every structure and every state(resp.fullpath)in we have(resp.).A state formula(resp.path formula)is satisfiable provided that for some structure and some states(resp.fullpath)in we have(resp.).In[17]we develop the algorithms for computing the accessibility relation and in[20]we implement them in NuSMV[2]in order to perform model checking.5.3.The logical analysis of a biological systemConsider the example of the interaction between the Virus and Macrophage discussed before.If the mathematical model chosen to describe the interaction is appropriate,then our system should have the property that,independently of the path of time that it will choose,always we will meet,in the future, the situation.Our logic allows us to formulate all these as a logical statement.We have:(5.1) For5.1we choose the urelements:for,for,for,for,for,for,forand for with.So,,,; is defined by:,,,,,,, and is defined by:is trueis trueis trueis trueis trueis trueis true The property we are interested in could be expressed as.It says that for all time paths exists at least a reachable state for which is a child of the master ambient,and are children of the Macrophage ambient .Further,for checking the truth value of this statement,a model checker could be used. Proving that our logical formula is true itfinally means that our mathematical model for describing our problem is a correct one.Vice versa,if is not valid,the model checker will give us a counter example that will show the conflict in our model.For such technics applied,the reader is referred to[20].6.ConclusionsThe logic we constructed on top of Ambient Calculus opens the perspective of using model checking algorithms(or software)developed for temporal logics in analyzing mobile computations.In this way we could predict the future of the systems(biological systems)described using the calculus.。
calculus
calculusCalculus: An Introduction to the Mathematical Study of ChangeIntroductionCalculus is a branch of mathematics that deals with the concept of change. It provides a powerful framework for understanding and analyzing a wide range of phenomena, from the motion of objects to the behavior of complex systems. In this document, we will explore the fundamentals of calculus, its applications, and some key concepts that underpin this fascinating field of study.History of CalculusThe origins of calculus can be traced back to ancient civilizations such as Babylon and Egypt, where rudimentary techniques for solving geometric problems and calculating areas and volumes were developed. However, it was in the 17th century that calculus took significant leaps forward with the contributions of two great mathematicians, Isaac Newton and Gottfried Wilhelm Leibniz.Newton and Leibniz independently developed a framework that allowed for the systematic treatment of rates of change and accumulation. Newton's approach, known as the method of fluxions, focused on the concept of instantaneous rates of change. Leibniz, on the other hand, introduced the notation we still use today, including the integral (∫) and the derivative (d/dd).Calculus ConceptsThe two main branches of calculus are differential calculus and integral calculus. Differential calculus focuses on the study of rates of change and slopes of curves, while integral calculus deals with the accumulation of quantities and the calculation of areas and volumes.Derivatives: The derivative of a function represents the rate at which the function is changing at any given point. It allows us to determine slopes of curves, maximal and minimal values, and even the behavior of functions on a larger scale. The derivative of a function is denoted by d′(d) or dd/dd.Integrals: Integrals provide a way to calculate the accumulation of quantities over intervals. They allow us to compute areas, volumes, and the total change of a function over a given interval. The integral of a function is denoted by ∫d(d)dd.Applications of CalculusCalculus has a wide range of applications in various fields, including physics, engineering, economics, and biology. Here are a few examples of how calculus is used in these disciplines:Physics: Calculus helps describe and predict the motion of objects, whether they are falling bodies, projectiles, or astronomical bodies. It also plays a crucial role in understanding concepts like acceleration, velocity, and force.Engineering: Engineers use calculus to analyze and design structures, control systems, and electrical circuits. Calculus is instrumental in optimizing systems by determining the optimal values of various parameters.Economics: Calculus is used in economics to model and analyze the behavior of markets, consumers, and producers. It provides a framework for understanding concepts such as supply and demand, marginal utility, and optimization of profit functions.Biology: Calculus is essential in mathematical modeling of biological processes, such as population growth, the spread of diseases, and the behavior of ecosystems. It enables researchers to make predictions and understand complex interactions within these systems.ConclusionCalculus is a fundamental branch of mathematics that revolutionized the way we understand and analyze change. Its concepts and principles have wide-ranging applications across many disciplines. From describing the motion of objects to modeling economic behavior and biological processes, calculus provides deep insights into the underlying mathematics governing these phenomena. As you continue your study of calculus, you will uncover even more fascinating applications and develop a deeper appreciation for its significance in the world around us.。
CLEP Calculus Exam Guide
18th EditionCalculusVisit our website at /clep for the most up-to-date information.The materials in these fi les are intended for personal use by students preparing for aCollege-Level Examination Program (CLEP®) examination. These materials are ownedand copyrighted by the College Board. All copyright notices must rem ain intact.Violations of this policy may be subject to legal action, including but not limited to,payment for each guide that is disseminated unlawfully and associated damages.© 2006 The College Board. All rights reserved. College Board, College-Level Examination Program, CLEP,CalculusDescription of the ExaminationThe Calculus examination covers skills and concepts that are usually taught in a one-semester college course in calculus. The content of each examination is approximately 60% limits and differential calculus and 40% integral calculus. Algebraic, trigonometric, exponential, logarithmic, and general functions are included. The exam is primarily concerned with an intuitive understanding of calculus and experience with its methods and applications. Knowledge of preparatory mathematics, including algebra, plane and solid geometry, trigonometry, and analytic geometry is assumed.Students are not permitted to use a calculator during the CLEP Calculus exam.The examination contains 45 questions to be answered in 90 minutes. Any time candidates spend on tutorials and providing personal information is in addition to the actual testing time.Knowledge and Skills RequiredQuestions on the exam require candidates to demon-strate the following abilities:• Solving routine problems involving the techniques of calculus (about 50% of the examination)• Solving nonroutine problems involving an understanding of the concepts and applications of calculus (about 50% of the examination)The subject matter of the calculus examination is drawn from the following topics. The percentages next to the main topics indicate the approximate percentages of exam questions on those topics. 5%Limits• Statement of properties, e.g., limit of a constant, sum, product, or quotient• Limits that involve infi nity, e.g., lim x x →01is nonexistent and lim sin x xx→∞=0• Continuity55% Differential CalculusThe Derivative • Defi nitions of the derivative,e.g., ′=−−→f a f x f a x ax a ()lim ()()and ′=+−→f x f x h f x hh ()lim()()0• Derivatives of elementary functions• Derivatives of sum, product, and quotient (including tan x and cot x )• Derivative of a composite function (chain rule), e.g., sin ,,ln()ax b ae kx kx +()• Derivative of an implicitly defi ned function • Derivative of the inverse of a function (including Arcsin x and Arctan x )• Derivatives of higher order• Corresponding characteristics of graphs of f f f ,,′′′and • Statement (without proof) of the Mean Value Theorem; applications and graphical illustrations• Relation between differentiability and continuity • Use of L ’Hôpital’s rule (quotient and indeterminate forms)Applications of the Derivative • Slope at a point• Tangent lines and linear approximation • Curve sketching: increasing and decreasing functions; relative andabsolute maximum and minimum points; concavity; points of infl ection • Extreme value problems• Velocity and acceleration of a particle moving along a line• Average and instantaneous rates of change • Related rates of change40% Integral CalculusAntiderivatives and Techniques of Integration • Concept of antiderivatives • Basic integration formulas• Integration by substitution (use of identi-ties, change of variable)The following sample questions do not appear on an actual CLEP examination. They are intendedto give potential test-takers an indication of the format and diffi culty level of the examination, and to provide content for practice and review. Knowing the correct answers to all of the sample questionsis not a guarantee of satisfactory performance on the exam.C A L C U L U S1. C2. E3. B4. D5. C6. D7. A8. D9. B10. D11. D12. D13. B14. D15. D16. B17. E18. B19. C20. E21. E22. C23. C 24. B25. D26. E27. B28. D29. C30. A31. B32. C33. D34. D35. B36. D37. C38. D39. C40. B41. B42. A43. B44. A45. AStudy ResourcesTo prepare for the Calculus exam, you shouldstudy the contents of at least one introductorycollege-level calculus textbook, which you can fi ndin most college bookstores. Y ou would do well toconsult several textbooks, because the approaches tocertain topics may vary. When selecting a textbook,check the table of contents against the “Knowledgeand Skills Required” for this exam.Additional suggestions for preparing for CLEP examsare given in “Preparing to Take CLEP Examinations.”Answer KeyI. Preparing to Take CLEP ExaminationsHaving made the decision to take one or more CLEP exams, most people then want to know how to prepare for them—how much, how long, when, and how should they go about it? The precise answers to these questions vary greatly from individual to individual. However, most candidates fi nd that some type of test preparation is helpful.Most people who take CLEP exams do so to show that they have already learned the key material taught in a college course. Many of them need only a quick review to assure themselves that they have not forgotten what they once studied, and to fi ll in some of the gaps in their knowledge of the subject. Others feel that they need a thorough review and spend several weeks studying for an exam. Some people take a CLEP exam as a kind of “fi nal exam” for independent study of a subject. This last group requires signifi cantly more study than do those who only need to review, and they may need some guidance from professors of the subjects they are s tudying.The key to how you prepare for CLEP exams often lies in locating those skills and areas of prior learning in which you are strong and deciding where to focus your energies. Some people may know a great deal about a certain subject area but may not test well. These individuals would probably be just as concerned about strengthening their test-taking skills as they would about studying for a specifi c test. Many mental and physical skills are used in preparing for a test. It is important not only to review or study for the exams but also to make certain that you are alert, relatively free of anxiety, and aware of how to approach standardized tests. Suggestions about developing test-taking skills and preparing psychologically and physically for a test are given in this chapter. The following section suggests ways of assessing your knowledge of the content of an exam and then reviewing and studying the material.Using the Examination GuidesWhether you are using the latest edition of this Study Guide, or you have downloaded an individual examination guide from the CLEP Web site, you will fi nd the same information. Each exam guide includes an outline of the knowledge and skills covered by the test, sample questions similar to those that appear on the exam, and tips for preparing to take the exam.You may also choose to contact a college in your area that offers a course with content comparable to that on the CLEP exam you want to take. If possible, use the textbook required for that course to help you prepare. To get this information, check the college’s catalog for a list of courses offered. Then call the admissions offi ce, e xplain what subject you’re interested in, and ask who in that academic department you can contact for specifi c information on textbooks and other study resources to use. Be sure that the college you’re interested in gives credit for the CLEP exam for which you’re preparing.Begin by carefully reading the test description and outline of knowledge and skills required for the exam in the exam guide. As you read through the topics listed, ask yourself how much you know about each one. Also note the terms, names, and symbols that are mentioned, and ask yourself whether you are familiar with them. This will give you a quick overview of how much you know about the subject. If you arefamiliar with nearly all the material, you will probably need a minimum of review; however, if topics and terms are unfamiliar, you will probably require substantial study to do well on the exam.If, after reviewing the test description provided in the exam guide, you fi nd that you need extensive review, put off answering the sample questions until you have done some reading in the subject. If you complete them before reviewing the material, you will probably look for the answers as you study, and they will not be a good assessment of your ability at a later date. Do not refer to the sample questions as you prepare for the exam. None of the sample questions appear on the CLEP exam, so concentrating on them without broader study of the subject won’t help you.If you think you are familiar with most of the test material, try to answer the sample questions, checking your responses against the answer key. Use the test-taking strategies described in the next chapter.Assessing Your Readiness for a CLEP ExaminationSuggestions for StudyingThe following suggestions have been gathered from people who have prepared for CLEP exams or other college-level tests.e CLEP tutorials.Make sure you are familiar with the computer-based format of the CLEP exams. Use the CLEPSampler, which can be downloaded from the CLEP Web site, to familiarize yourself with CLEP CBT exams before taking the test; it’s also the only offi cial CLEP tutorial program for computer-based testing. You can fi nd the Sampler on the Web at /clep. If you are not comfortable using a computer, you can practice the necessary pointing, clicking, and scrolling skills by working with the Sampler. You’ll also be able to practice using the testing tools that will help you navigate throughout the test, and you’ll see the types of questions you’ll be required to answer.If you don’t have access to a computer, check with the library or test center at the school where you’ll be testing. Many CLEP test centers and college libraries will have the Sampler installed on computers in public areas, so you’ll be able to practice and review before your test date. The tutorials are also part of the testing software, and you’ll be able to work through them before you begin your test.Check with the test center to see how much time will be allotted for your testing appointment; then you can determine how much time you might need to spend on the tutorials.Remember, if you want to review content covered by each examination, the exam description includes a content outline, a description of the knowledge and skills required to do well, and sample questions. An answer key is also included. However, th is exam guide is not intended to replace a textbook. Additional study may be required.2.Defi ne your goals and locate study materials.First, determine your study goals. Set aside a block of time to review the exam guide andthen decide which exam(s) you will take. Using the guidelines for knowledge and skillsrequired, locate suitable resource materials. If a preparation course is offered by an adult school or college in your area, you might fi nd it helpful to enroll. (You should be aware, however, that such courses are not authorized or sponsored by the College Board. The College Board has no responsibility for the content of these courses; nor are they responsible for books on preparing for CLEP exams that have been published by other organizations.) If you know others who have taken CLEP exams, ask them how they prepared.You may want to get a copy of a syllabus for the college course that is comparable to the CLEP exam(s) you plan to take. Some colleges, like MIT and Carnegie Mellon, offer their course materials for free online; these can be an excellent resource. You can also ask the appropriate professor at the school you’ll be attending, or check his or her Web site, for a reading list. Use the syllabus, course materials and/or reading list as your guide for selecting textbooks and study materials. You may purchase these or check them out of your local library. Educational Web sites, like those offered by PBS or the National Geographic Society, can be helpful as well.Check with your librarian about locating study aids relevant to the exams you plan to take. These supplementary materials may include, for example, videos or DVDs made by education-oriented companies and organizations; language tapes; and computer software. And don’t forget that what you do with your leisure time can be very educational, whether it’s surfi ng current-events Web sites,watching a PBS series, reading a fi nancial newsletter, or attending a play.3.Find a good place to study.To determine what kind of place you need for studying, ask yourself these questions: Do I need a quiet place? Does the telephone distract me? Do objects I see in this place remind me of things I should do?Is it too warm? Is it well lit? Am I too comfortable here? Do I have space to spread out my materials?You may fi nd the library more conducive to studying than your home. If you decide to study at home or in your dorm, you might prevent interruptions by other household members by putting a sign on the door of your study room to indicate when you will be available.4.Schedule time to study.To help you determine where studying best fi ts into your schedule, try this exercise: Make a list of your daily activities (for example, sleeping, working, eating, attending class, sports, or exercise) and estimate how many hours a day you spend on each activity. Now, rate all the activities on your list in order of their importance and evaluate your use of time. Often people are astonished at how an average day appears from this perspective. You may discover that your time can be scheduled in alternative ways.For example, you could remove the least important activities from your day and devote that time to studying or to another important activity.5.Establish a study routine and a set of goals.To study effectively, you should establish specifi c goals and a schedule for accomplishing them. Some people fi nd it helpful to write out a weekly schedule and cross out each study period when it iscompleted. Others maintain their concentration better by writing down the time when they expect to complete a study task. Most people fi nd short periods of intense study more productive than long stretches of time. For example, they may follow a regular schedule of several 20- or 30-minute study periods with short breaks between them. Some people like to allow themselves rewards as theycomplete each study goal. It is not essential that you accomplish every goal exactly within yourschedule; the point is to be committed to your task.6.Learn how to take an active role in studying.If you have not done much studying for some time, you may fi nd it diffi cult to concentrate at fi rst. Trya method of studying, such as the one outlined below, that will help you concentrate on and rememberwhat you read.a.First, read the chapter summary and the introduction so you will know what to look for inyour reading.b.Next, convert the section or paragraph headlines into questions. For example, if you are reading asection entitled “The Causes of the American Revolution,” ask yourself, “What were the causes of the American Revolution?” Compose the answer as you read the paragraph. Reading and answering questions aloud will help you understand and remember the material.c. Take notes on key ideas or concepts as you read. Writing will also help you fi x concepts more fi rmlyin your mind. Underlining key ideas or writing notes in your book can be helpful and will be useful for review. Underline only important points. If you underline more than a third of each paragraph, you are probably underlining too much.d.If there are questions or problems at the end of a chapter, answer or solve them on paper as if youwere asked to do them for homework. Mathematics textbooks (and some other books) sometimes include answers to some or all of the exercises. If you have such a book, write your answers before looking at the ones given. When problem solving is involved, work enough problems to master the required methods and concepts. If you have diffi culty with problems, review any sample problems or explanations in the chapter.e.To retain knowledge, most people have to review the material periodically. If you are preparing foran exam over an extended period of time, review key concepts and notes each week or so. Do not wait for weeks to review the material or you will need to relearn much of it.Psychological and Physical PreparationMost people feel at least some nervousness before taking a test. Adults who are returning to college may not have taken tests in many years, or they may have had little experience with standardized tests. Some younger students, as well, are uncomfortable with testing situations. People who received their education in countries outside the United States may fi nd that many tests given in this country are quite different from the ones they are accustomed to taking.Not only might candidates fi nd the types of tests and questions unfamiliar, but other aspects of the testing environment may be strange as well. The physical and mental stress that results from meeting this new experience can hinder a candidate’s ability to demonstrate his or her true degree of knowledge in the subject area being tested. For this reason, it is important to go to the test center well prepared, both mentally and physically, for taking the test. You may fi nd the following suggestions helpful.1.Familiarize yourself as much as possible with the test and the test situation before the day of the exam.It will be helpful for you to know ahead of time:a.How much time will be allowed for the test and whether there are timed subsections. (Thisinformation is included in the examination guides and in the CLEP Sampler.)b.What types of questions and directions appear on the exam. (See the examination guides and theCLEP Sampler.)c.How your test score will be computed.d. In which building and room the exam will be administered. If you don’t know where the building is,get directions ahead of time.e.The time of the test administration. You may wish to confi rm this information a day or two before theexam and fi nd out what time the building and room will be open so that you can plan to arrive early.f. Where to park your car and whether you will need a parking permit or, if you will be taking publictransportation, which bus or train to take and the location of the nearest stop.g.Whether there will be a break between exams (if you will be taking more than one on the same day),and whether there is a place nearby where you can get something to eat or drink.2.Be relaxed and alert while you are taking the exam:a.Get a good night’s sleep. Last-minute cramming, particularly late the night before, is usuallycounterproductive.b.Eat normally. It is usually not wise to skip breakfast or lunch on the day you take the exam or to eat abig meal just before testing.c.Avoid tranquilizers and stimulants. If you follow the other directions in this book, you won’t needartifi cial aids. It’s better to be a little tense than to be drowsy, but stimulants such as coffee and cola can make you nervous and interfere with your concentration.d.Don’t drink a lot of liquids before taking the exam. Leaving to use the restroom during testing willdisturb your concentration and reduce the time you have to complete the exam.e.If you are inclined to be nervous or tense, learn some relaxation exercises and use them to preparefor the exam.3. Be sure to:a.Arrive early enough so that you can fi nd a parking place, locate the test center, and get settledcomfortably before testing begins. Allow some extra time in case you are delayed unexpectedly.b.Take the following with you:●Any registration forms or printouts required by the test center. Make sure you have fi lled out allnecessary paperwork in advance of your testing date.●Your driver’s license, passport, or other government-issued identifi cation that includes yourphotograph and signature, as well as a secondary form of ID that includes a photo and/or yoursignature, such as a student ID, military ID, social security card, or credit card. You will be asked to show this identifi cation to be admitted to the testing area.● A valid credit card to pay the $60 examination fee. (This fee is subject to change.) Although acredit card is the preferred method of payment, you can also pay by check or money order(payable to the College-Level Examination Program). Your test center may require an additionaladministration fee. Contact the test center to determine the amount and the method of payment.●Two pencils with good erasers. You may need a pencil for writing an outline or fi guring out mathproblems. Mechanical pencils are prohibited in the testing room.●Your glasses if you need them for reading or seeing the chalkboard or wall clock.c.Leave all books, papers, and notes outside the test center. You will not be permitted to use your ownscratch paper; it will be provided by the test center.d.Do not take a calculator to the exam. If a calculator is required, it will be built into the testingsoftware and available to you on the computer. The CLEP Sampler and the pretest tutorials willshow you how to use that feature.e.Do not bring a cell phone or other electronic devices into the testing room.f.Be prepared to adjust to an uncomfortable temperature in the testing room. Wear layers of clothingthat can be removed if the room is too hot but that will keep you warm if it is too cold.4.When you enter the test room:a.Although you will be assigned to a computer testing station, the test center administrator can usuallyaccommodate special needs. Be sure to communicate your needs before the day you test.b. Read directions carefully and listen to all instructions given by the test administrator. If you don’tunderstand the directions, ask for help before test timing begins. If you must ask a question aftertesting has begun, raise your hand and a proctor will assist you. The proctor can answer certain kinds of questions but cannot help you with the exam.c.Know your rights as a test-taker. You can expect to be given the full working time allowed for takingthe exam and a reasonably quiet and comfortable place in which to work. If a poor testing situation is preventing you from doing your best, ask whether the situation can be remedied. If bad testingconditions cannot be remedied, ask the person in charge to report the problem on an ElectronicIrregularity Report that will be submitted with your test results. You may also wish to immediately write a letter to CLEP, P.O. Box 6656, Princeton, NJ 08541-6656. Describe the exact circumstances as completely as you can. Be sure to include the name of the test center, the test date, and the name(s) of the exam(s) you took. The problem will be investigated to make sure it does not happen again, and, if the problem is serious enough, arrangements will be made for you to retake the exam without charge.Arrangements for Students with DisabilitiesCLEP is committed to working with test-takers with disabilities. If you have a learning or physical disability that would prevent you from taking a CLEP exam under standard conditions, you may request special accommodations and arrangements to take it on a regularly scheduled test date or at a special administration. Contact a CLEP test center prior to registration about testing accommodations and to ensure the accommodation you are requesting is available. Each test center sets its own guidelines in terms of deadlines for submission of documentation and approval of accommodations. Only students with documented hearing, learning, physical, or visual disabilities are eligible to receive testing accommodations. Also, it is important to ensure that you are taking the exam(s) with accommodations that are approved by your score recipient institution.Testing accommodations that may be provided with appropriate disability documentation include:●ZoomText (screen magnifi cation)● Modifi able screen colors●Use of a reader or amanuensis or sign language interpreter● Extended time● Untimed rest breaksII. Taking the ExaminationsA person may know a great deal about the subject being tested but not be able to demonstrate it on the exam. Knowing how to approach an exam is an important part of the testing process. While a command of test-taking skills cannot substitute for knowledge of the subject matter, it can be a signifi cant factor in successful testing.Test-taking skills enable a person to use all available information to earn a score that truly refl ects her or his ability. There are different strategies for approaching different kinds of exam questions. For example, free-response and multiple-choice questions require very different approaches. Other factors, such as how the exam will be graded, may also infl uence your approach to the exam and your use of test time. Thus, your preparation for an exam should include fi nding out all you can about the exam so you can use the most effective test-taking strategies.Taking CLEP Exams1.Listen carefully to any instructions given by the test administrator and read the on-screen instructionsbefore you begin to answer the questions.2.Keep an eye on the clock and the timing that is built into the testing software. You have the optionof turning the clock on or off at any time. As you proceed, make sure that you are not working too slowly. You should have answered at least half the questions in a section when half the time forthat section has passed. If you have not reached that point in the section, speed up your pace on the remaining questions.3.Before answering a question, read the entire question, including all the answer choices. Don’t thinkthat because the fi rst or second answer choice looks good to you, it isn’t necessary to read ther emaining options. Instructions usually tell you to select the “best’’ answer. Sometimes one answer choice is partially correct but a nother option is better; therefore, it’s usually a good idea to read all the answers before you choose one.4.Read and consider every question. Questions that look complicated at fi rst glance may not actually beso diffi cult once you have read them carefully.5.Do not puzzle too long over any one question. If you don’t know the answer after you’ve considered itbriefl y, go on to the next question. Mark that question using the mark tool at the bottom of the screen, and go back to review the question later, if you have time.6.Watch for the following key words in test questions:all generally never perhapsalways however none rarelybut may not seldomexcept must often sometimesevery necessary only usuallyWhen a question or answer option contains words such as ‘‘always,’’ ‘‘every,’’ ‘‘only,’’ ‘‘never,’’ and “none,” there can be no exceptions to the answer you choose. Use of words such as ‘‘often,’’ “rarely,”‘‘sometimes,’’ and ‘‘generally’’ indicates that there may be some exceptions to the answer.7.Make educated guesses. There is no penalty for incorrect answers. It is to your benefi t to guess if youdo not know an answer since CLEP CBT uses “rights-only” scoring. (An explanation of theprocedures used for scoring CLEP exams is given in the next chapter.) If you are not sure of thecorrect answer but have some knowledge of the question and are able to eliminate one or more of the answer choices as wrong, your chance of getting the right answer is improved.8.Do not waste your time looking for clues to right answers based on fl aws in question wording orpatterns in correct answers. CLEP puts a great deal of effort into developing valid, reliable, and fair exams. CLEP test development committees are composed of college faculty who are experts in the subjects covered by the exams and are appointed by the College Board to write test questions and to scrutinize each question that is included on a CLEP exam. Faculty committee members make every effort to ensure that the questions are not ambiguous, that they have only one correct answer, and that they cover college-level topics. These committees do not intentionally include ‘‘trick’’ questions. If you think a question is fl awed, ask the test administrator to report it, or write immediately to CLEP Test Development, P.O. Box 6600, Princeton, NJ 08541-6600. Include the name of the exam and test center, the exam date, and the number of the exam question. All such inquiries are investigated by testdevelopment professionals.。
十八世纪的数学发展【英文】
Geometry
Euler (1765) showed that in any triangle, the orthocenter, circumcenter, centroid, and ninepoint center are collinear. Because of that the line which connects the points above is called Euler line.
GABRIEL CRAMER
GABRIEL CRAMER (1704-1752)
Swiss mathematician He give the solution of St. Peterburg paradox He worked on analysis and determinants He is the most famous by his rule (Cramer’s rule) which gives a solution of a system of linear equations using determinants.
THOMAS SIMPSON (1710-1761)
JEAN LE ROND D’ALAMBERT
JEAN LE ROND D’ALAMBERT (1717-1783)
He dealt with problems of dinamics and fluids and especially with problem of vibrating string which leads to solving partial diferential equations During his second part of life, he was mainly occupied with the great French encyclopedia
David Hilbert - Mathematical Problems
Mathematical ProblemsLecture delivered before the International Congress ofMathematicians at Paris in 1900By Professor David Hilbert1Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future. The deep significance of certain problems for the advance of mathematical science in general and the important role which they play in the work of the individual investigator are not to be denied. As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research requires its problems. It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.It is difficult and often impossible to judge the value of a problem correctly in advance; for the final award depends upon the gain which science obtains from the problem. Nevertheless we can ask whether there are general criteria which mark a good mathematical problem. An old French mathematician said: "A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street." This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us.Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.The mathematicians of past centuries were accustomed to devote themselves to the solution of difficult particular problems with passionate zeal. They knew the value of difficult problems. I remind you only of the "problem of the line of quickest descent," proposed by John Bernoulli. Experience teaches, explains Bernoulli in the public announcement of this problem, that lofty minds are led to strive for the advance of science by nothing more than by laying before them difficult and at the same time useful problems, and he therefore hopes to earn the thanks of the mathematical world by following the example of men like Mersenne, Pascal, Fermat, Viviani and others and laying before the distinguished analysts of his time a problem by which, as a touchstone, they may test the value of their methods and measure their strength. The calculus of variations owes its origin to this problem of Bernoulli and to similar problems.Fermat had asserted, as is well known, that the diophantine equationx n + y n = z n(x, y and z integers) is unsolvable—except in certain self evident cases. The attempt to prove this impossibility offers a striking example of the inspiring effect which such a very special and apparently unimportant problem may have upon science. For Kummer, incited by Fermat's problem, was led to the introduction of ideal numbers and to the discovery of the law of the unique decomposition of the numbers of a circular field into ideal prime factors—a law which today, in its generalization to any algebraic field by Dedekind and Kronecker, stands at the center of the modern theory of numbers and whose significance extends far beyond the boundaries of number theory into the realm of algebra and the theory of functions.To speak of a very different region of research, I remind you of the problem of three bodies. The fruitful methods and the far-reaching principles which Poincaré has brought into celestial mechanics and which are today recognized and applied in practical astronomy are due to the circumstance that he undertook to treat anew that difficult problem and to approach nearer a solution.The two last mentioned problems—that of Fermat and the problem of the three bodies—seem to us almost like opposite poles—the former a free invention of pure reason, belonging to the region of abstract number theory, the latter forced upon us by astronomy and necessary to an understanding of the simplest fundamental phenomena of nature.But it often happens also that the same special problem finds application in the most unlike branches of mathematical knowledge. So, for example, the problem of the shortest line plays a chief and historically important part in the foundations of geometry, in the theory of curved lines and surfaces, in mechanics and in the calculus of variations. And how convincingly has F. Klein, in his work on the icosahedron, pictured the significance which attaches to the problem of the regular polyhedra in elementary geometry, in group theory, in the theory of equations and in that of linear differential equations.In order to throw light on the importance of certain problems, I may also refer to Weierstrass, who spokeof it as his happy fortune that he found at the outset of his scientific career a problem so important as Jacobi's problem of inversion on which to work.Having now recalled to mind the general importance of problems in mathematics, let us turn to the question from what sources this science derives its problems. Surely the first and oldest problems in every branch of mathematics spring from experience and are suggested by the world of external phenomena. Even the rules of calculation with integers must have been discovered in this fashion in a lower stage of human civilization, just as the child of today learns the application of these laws by empirical methods. The same is true of the first problems of geometry, the problems bequeathed us by antiquity, such as the duplication of the cube, the squaring of the circle; also the oldest problems in the theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory of potential—to say nothing of the further abundance of problems properly belonging to mechanics, astronomy and physics. But, in the further development of a branch of mathematics, the human mind, encouraged by the success of its solutions, becomes conscious of its independence. It evolves from itself alone, often without appreciable influence from without, by means of logical combination, generalization, specialization, by separating and collecting ideas in fortunate ways, new and fruitful problems, and appears then itself as the real questioner. Thus arose the problem of prime numbers and the other problems of number theory, Galois's theory of equations, the theory of algebraic invariants, the theory of abelian and automorphic functions; indeed almost all the nicer questions of modern arithmetic and function theory arise in this way.In the meantime, while the creative power of pure reason is at work, the outer world again comes into play, forces upon us new questions from actual experience, opens up new branches of mathematics, and while we seek to conquer these new fields of knowledge for the realm of pure thought, we often find the answers to old unsolved problems and thus at the same time advance most successfully the old theories. And it seems to me that the numerous and surprising analogies and that apparently prearranged harmony which the mathematician so often perceives in the questions, methods and ideas of the various branches of his science, have their origin in this ever-recurring interplay between thought and experience.It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning. Indeed the requirement of rigor, which has become proverbial in mathematics, corresponds to a universal philosophical necessity of our understanding; and, on the other hand, only by satisfying this requirement do the thought content and the suggestiveness of the problem attain their full effect. A new problem, especially when it comes from the world of outer experience, is like a young twig, which thrives and bears fruit only when it is grafted carefully and in accordance with strict horticultural rules upon the old stem, the established achievements of our mathematical science.Besides it is an error to believe that rigor in the proof is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigor forces us to find out simpler methods of proof. It also frequently leads the way to methods which are more capable of development than the old methods of less rigor. Thus the theory of algebraic curves experienced a considerable simplification and attained greater unity by means of the more rigorous function-theoretical methods and the consistent introduction of transcendental devices. Further, the proof that the power series permits the application of the four elementary arithmetical operations as well as the term by term differentiation and integration, and the recognition of the utility of the power series depending upon this proof contributed materially to the simplification of all analysis, particularly of the theory of elimination and the theory of differential equations, and also of the existence proofs demanded in those theories. But the most striking example for my statement is the calculus of variations. The treatment of the first and second variations of definite integrals required in part extremely complicated calculations, and the processes applied by the old mathematicians had not the needful rigor. Weierstrass showed us the way to a new and sure foundation of the calculus of variations. By the examples of the simple and double integral I will show briefly, at the close of my lecture, how this way leads at once to a surprising simplification of the calculus of variations. For in the demonstration of the necessary and sufficient criteria for the occurrence of a maximum and minimum, the calculation of the second variation and in part, indeed, the wearisome reasoning connected with the first variation may be completely dispensed with—to say nothing of the advance which is involved in the removal of the restriction to variations for which the differential coefficients of the function vary but slightly.While insisting on rigor in the proof as a requirement for a perfect solution of a problem, I should like, on the other hand, to oppose the opinion that only the concepts of analysis, or even those of arithmetic alone, are susceptible of a fully rigorous treatment. This opinion, occasionally advocated by eminent men, I consider entirely erroneous. Such a one-sided interpretation of the requirement of rigor would soon lead to the ignoring of all concepts arising from geometry, mechanics and physics, to a stoppage of the flow of new material from the outside world, and finally, indeed, as a last consequence, to the rejection of the ideas of the continuum and of the irrational number. But what an important nerve, vital to mathematical science, would be cut by the extirpation of geometry and mathematical physics! On the contrary I think that wherever, from the side of the theory of knowledge or in geometry, or from the theories of natural or physical science, mathematical ideas come up, the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms, that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts.To new concepts correspond, necessarily, new signs. These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts. So the geometrical figures are signs or mnemonic symbols of space intuition and are used as such by all mathematicians. Who does not always use along with the double inequality a > b > c the picture of three points following one another on a straight line as the geometrical picture of the idea "between"? Who does not make use of drawings of segments and rectangles enclosed in one another, when it is required to prove with perfect rigor a difficult theorem on the continuity of functions or the existence of points of condensation? Whocould dispense with the figure of the triangle, the circle with its center, or with the cross of three perpendicular axes? Or who would give up the representation of the vector field, or the picture of a family of curves or surfaces with its envelope which plays so important a part in differential geometry, in the theory of differential equations, in the foundation of the calculus of variations and in other purely mathematical sciences?The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas; and no mathematician could spare these graphic formulas, any more than in calculation the insertion and removal of parentheses or the use of other analytical signs.The use of geometrical signs as a means of strict proof presupposes the exact knowledge and complete mastery of the axioms which underlie those figures; and in order that these geometrical figures may be incorporated in the general treasure of mathematical signs, there is necessary a rigorous axiomatic investigation of their conceptual content. Just as in adding two numbers, one must place the digits under each other in the right order, so that only the rules of calculation, i. e., the axioms of arithmetic, determine the correct use of the digits, so the use of geometrical signs is determined by the axioms of geometrical concepts and their combinations.The agreement between geometrical and arithmetical thought is shown also in that we do not habitually follow the chain of reasoning back to the axioms in arithmetical, any more than in geometrical discussions. On the contrary we apply, especially in first attacking a problem, a rapid, unconscious, not absolutely sure combination, trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols, which we could dispense with as little in arithmetic as with the geometrical imagination in geometry. As an example of an arithmetical theory operating rigorously with geometrical ideas and signs, I may mention Minkowski's work, Die Geometrie der Zahlen.2Some remarks upon the difficulties which mathematical problems may offer, and the means of surmounting them, may be in place here.If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standpoint, not only is this problem frequently more accessible to our investigation, but at the same time we come into possession of a method which is applicable also to related problems. The introduction of complex paths of integration by Cauchy and of the notion of the IDEALS in number theory by Kummer may serve as examples. This way for finding general methods is certainly the most practicable and the most certain; for he who seeks for methods without having a definite problem in mind seeks for the most part in vain.In dealing with mathematical problems, specialization plays, as I believe, a still more important part than generalization. Perhaps in most cases where we seek in vain the answer to a question, the cause of the failure lies in the fact that problems simpler and easier than the one in hand have been either not at all or incompletely solved. All depends, then, on finding out these easier problems, and on solving them bymeans of devices as perfect as possible and of concepts capable of generalization. This rule is one of the most important levers for overcoming mathematical difficulties and it seems to me that it is used almost always, though perhaps unconsciously.Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed. The problem then arises: to show the impossibility of the solution under the given hypotheses, or in the sense contemplated. Such proofs of impossibility were effected by the ancients, for instance when they showed that the ratio of the hypotenuse to the side of an isosceles right triangle is irrational. In later mathematics, the question as to the impossibility of certain solutions plays a preeminent part, and we perceive in this way that old and difficult problems, such as the proof of the axiom of parallels, the squaring of the circle, or the solution of equations of the fifth degree by radicals have finally found fully satisfactory and rigorous solutions, although in another sense than that originally intended. It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts. Take any definite unsolved problem, such as the question as to the irrationality of the Euler-Mascheroni constant C, or the existence of an infinite number of prime numbers of the form 2n + 1. However unapproachable these problems may seem to us and however helpless we stand before them, we have, nevertheless, the firm conviction that their solution must follow by a finite number of purely logical processes.Is this axiom of the solvability of every problem a peculiarity characteristic of mathematical thought alone, or is it possibly a general law inherent in the nature of the mind, that all questions which it asks must be answerable? For in other sciences also one meets old problems which have been settled in a manner most satisfactory and most useful to science by the proof of their impossibility. I instance the problem of perpetual motion. After seeking in vain for the construction of a perpetual motion machine, the relations were investigated which must subsist between the forces of nature if such a machine is to be impossible;3 and this inverted question led to the discovery of the law of the conservation of energy, which, again, explained the impossibility of perpetual motion in the sense originally intended.This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.The supply of problems in mathematics is inexhaustible, and as soon as one problem is solved numerous others come forth in its place. Permit me in the following, tentatively as it were, to mention particular definite problems, drawn from various branches of mathematics, from the discussion of which an advancement of science may be expected.Let us look at the principles of analysis and geometry. The most suggestive and notable achievements of the last century in this field are, as it seems to me, the arithmetical formulation of the concept of thecontinuum in the works of Cauchy, Bolzano and Cantor, and the discovery of non-euclidean geometry by Gauss, Bolyai, and Lobachevsky. I therefore first direct your attention to some problems belonging to these fields.1. Cantor's problem of the cardinal number of the continuumTwo systems, i. e, two assemblages of ordinary real numbers or points, are said to be (according to Cantor) equivalent or of equal cardinal number, if they can be brought into a relation to one another such that to every number of the one assemblage corresponds one and only one definite number of the other. The investigations of Cantor on such assemblages of points suggest a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving. This is the theorem:Every system of infinitely many real numbers, i. e., every assemblage of numbers (or points), is either equivalent to the assemblage of natural integers, 1, 2, 3,... or to the assemblage of all real numbers and therefore to the continuum, that is, to the points of a line; as regards equivalence there are, therefore, only two assemblages of numbers, the countable assemblage and the continuum.From this theorem it would follow at once that the continuum has the next cardinal number beyond that of the countable assemblage; the proof of this theorem would, therefore, form a new bridge between the countable assemblage and the continuum.Let me mention another very remarkable statement of Cantor's which stands in the closest connection with the theorem mentioned and which, perhaps, offers the key to its proof. Any system of real numbers is said to be ordered, if for every two numbers of the system it is determined which one is the earlier and which the later, and if at the same time this determination is of such a kind that, if a is before b and b is before c, then a always comes before c. The natural arrangement of numbers of a system is defined to be that in which the smaller precedes the larger. But there are, as is easily seen infinitely many other ways in which the numbers of a system may be arranged.If we think of a definite arrangement of numbers and select from them a particular system of these numbers, a so-called partial system or assemblage, this partial system will also prove to be ordered. Now Cantor considers a particular kind of ordered assemblage which he designates as a well ordered assemblage and which is characterized in this way, that not only in the assemblage itself but also in every partial assemblage there exists a first number. The system of integers 1, 2, 3, ... in their natural order is evidently a well ordered assemblage. On the other hand the system of all real numbers, i. e., the continuum in its natural order, is evidently not well ordered. For, if we think of the points of a segment of a straight line, with its initial point excluded, as our partial assemblage, it will have no first element.The question now arises whether the totality of all numbers may not be arranged in another manner so that every partial assemblage may have a first element, i. e., whether the continuum cannot be consideredas a well ordered assemblage—a question which Cantor thinks must be answered in the affirmative. It appears to me most desirable to obtain a direct proof of this remarkable statement of Cantor's, perhaps by actually giving an arrangement of numbers such that in every partial system a first number can be pointed out.2. The compatibility of the arithmetical axiomsWhen we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps. Upon closer consideration the question arises: Whether, in any way, certain statements of single axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another. But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results.In geometry, the proof of the compatibility of the axioms can be effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms. Any contradiction in the deductions from the geometrical axioms must thereupon be recognizable in the arithmetic of this field of numbers. In this way the desired proof for the compatibility of the geometrical axioms is made to depend upon the theorem of the compatibility of the arithmetical axioms.On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms. The axioms of arithmetic are essentially nothing else than the known rules of calculation, with the addition of the axiom of continuity. I recently collected them4 and in so doing replaced the axiom of continuity by two simpler axioms, namely, the well-known axiom of Archimedes, and a new axiom essentially as follows: that numbers form a system of things which is capable of no further extension, as long as all the other axioms hold (axiom of completeness). I am convinced that it must be possible to find a direct proof for the compatibility of the arithmetical axioms, by means of a careful study and suitable modification of the known methods of reasoning in the theory of irrational numbers.To show the significance of the problem from another point of view, I add the following observation: If contradictory attributes be assigned to a concept, I say, that mathematically the concept does not exist. So, for example, a real number whose square is -l does not exist mathematically. But if it can be proved that the attributes assigned to the concept can never lead to a contradiction by the application of a finite number of logical processes, I say that the mathematical existence of the concept (for example, of a number or a function which satisfies certain conditions) is thereby proved. In the case before us, wherewe are concerned with the axioms of real numbers in arithmetic, the proof of the compatibility of the axioms is at the same time the proof of the mathematical existence of the complete system of real numbers or of the continuum. Indeed, when the proof for the compatibility of the axioms shall be fully accomplished, the doubts which have been expressed occasionally as to the existence of the complete system of real numbers will become totally groundless. The totality of real numbers, i. e., the continuum according to the point of view just indicated, is not the totality of all possible series in decimal fractions, or of all possible laws according to which the elements of a fundamental sequence may proceed. It is rather a system of things whose mutual relations are governed by the axioms set up and for which all propositions, and only those, are true which can be derived from the axioms by a finite number of logical processes. In my opinion, the concept of the continuum is strictly logically tenable in this sense only. It seems to me, indeed, that this corresponds best also to what experience and intuition tell us. The concept of the continuum or even that of the system of all functions exists, then, in exactly the same sense as the system of integral, rational numbers, for example, or as Cantor's higher classes of numbers and cardinal numbers. For I am convinced that the existence of the latter, just as that of the continuum, can be proved in the sense I have described; unlike the system of all cardinal numbers or of all Cantor s alephs, for which, as may be shown, a system of axioms, compatible in my sense, cannot be set up. Either of these systems is, therefore, according to my terminology, mathematically non-existent.From the field of the foundations of geometry I should like to mention the following problem:3. The equality of two volumes of two tetrahedra of equal bases and equal altitudesIn two letters to Gerling, Gauss5 expresses his regret that certain theorems of solid geometry depend upon the method of exhaustion, i. e., in modern phraseology, upon the axiom of continuity (or upon the axiom of Archimedes). Gauss mentions in particular the theorem of Euclid, that triangular pyramids of equal altitudes are to each other as their bases. Now the analogous problem in the plane has been solved.6 Gerling also succeeded in proving the equality of volume of symmetrical polyhedra by dividing them into congruent parts. Nevertheless, it seems to me probable that a general proof of this kind for the theorem of Euclid just mentioned is impossible, and it should be our task to give a rigorous proof of its impossibility. This would be obtained, as soon as we succeeded in specifying two tetrahedra of equal bases and equal altitudes which can in no way be split up into congruent tetrahedra, and which cannot be combined with congruent tetrahedra to form two polyhedra which themselves could be split up into congruent tetrahedra.74. Problem of the straight line as the shortest distance between two pointsAnother problem relating to the foundations of geometry is this: If from among the axioms necessary to。
mathematical problems in engineering 特短的文章
mathematical problems in engineering特短的文章Mathematics has long been recognized as one of the most important fields of study, and its applications can be seenin a variety of areas including engineering. In this article, we will discuss some common mathematical problems in engineering and their significance.Step 1: Calculus ProblemsCalculus is an essential branch of mathematics that has vast applications in engineering. One of the most common problems in engineering that require calculus is optimization. Engineers often use calculus to find the optimum value of a particular variable that can lead to an optimal solution for a given problem. Calculus is also used for modeling complex engineering problems that involve multidimensional variables, such as heat transfer, fluid mechanics, and structural analysis.Step 2: Matrix ProblemsMatrices are a crucial concept in linear algebra and have vast applications in engineering. Matrices are used to model systems of equations and their solutions. Engineers often use matrices to solve problems related to dynamic systems,control systems, and system analysis. Moreover, matrix problems are essential in image processing, optimization, and data analysis.Step 3: Differential EquationsDifferential equations are a mathematical tool used to modelthe rate of change of certain variables in a system over time. Engineers often use differential equations to solve problems related to fluid dynamics, mechanics, and electrical engineering. Differential equations are also important in modeling and understanding the behavior of circuits, systems, and machinery.Step 4: Probability and StatisticsProbability and statistics are essential branches of mathematics used in engineering. Engineers use probabilityand statistics to quantify the uncertainty and variability associated with measurement data and designs. Applications of probability and statistics in engineering include reliability and quality control, risk analysis, and experimental design.In conclusion, mathematical problems play a criticalrole in engineering. Calculus problems are prevalent in optimization, matrix problems in the modeling of systems, differential equations for modeling rates of change over time, and probability and statistics for quantifying uncertainties. Engineers use mathematical problem-solving skills to solve challenging problems in diverse fields such as electrical, mechanical, aerospace, civil, and chemical engineering. As technology evolves, mathematical problems in engineering will continue to become more challenging, and engineers must be prepared to take on these challenges.。
三年级微积分的计算法则英语阅读理解25题
三年级微积分的计算法则英语阅读理解25题1<背景文章>Calculus is a very interesting subject. What is calculus? Calculus is a branch of mathematics that deals with rates of change and accumulation. For example, when we want to know how fast a car is going, we can use calculus. The basic concept of calculus is the derivative. The derivative tells us how fast something is changing at a particular moment. For instance, if we have a function that represents the position of a car over time, the derivative of that function gives us the speed of the car at any given time.Now let's talk about another important concept in calculus, the integral. The integral is used to find the total amount of something. For example, if we know the speed of a car at different times, we can use the integral to find the total distance traveled by the car.Calculus can be a bit difficult to understand at first, but with practice, it can become easier.1. What is calculus a branch of?A. PhysicsB. ChemistryC. MathematicsD. Biology答案：C。
谴责申明：世界十大数学难题的解答,圆球体层式的解答。解读
标题：世界十大数学难题的解答.圆球体层式解答.作者: 百度里的昵称“蔡於竟道”Apology statement.Condemn stated: relatively speaking, this layer ball type theory.Is true, and I in baidu's nickname, "CAI in" registration in May 2014, at the same time.Can other will not.B: yes.Other, opposite in other forms.0, 7 years, told the diameter of the sphere, and how the score, N, anything better to do one thing at the time, including the universe at the time, and so on.Principle of outer principle, principle of the inner principle.About is like a thing well done.The condemned man: baidu's nickname, "CAI YU JING DAO ".Taizhou city, zhejiang province.China.In October 2015. 3. (machine translation, I do not know right?)道歉申明。
谴责申明：相对来说，此圆球体层式理论。
真正的，是和本人在百度的昵称“蔡於竟道”，注册时2014年5月，同时发表的。
其他的可以都将不算。
是的。
其他的，相对是以其他形式讲过。
零七年时，讲过的0，圆球体的直径，怎样的参比数，N，任何当时不如做一件事，包括宇宙当时，等。
ap-2002-calculus-bc-free-response-questions-form-b
3
2002 AP® CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)
CALCULUS BC SECTION II, Part B
Time—45 minutes Number of problems—3
No calculator is allowed for these problems.
(c) Write an expression involving one or more integrals that gives the perimeter of R. Do not evaluate.
END OF PART A OF SECTI by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.
(d) Is the distance traveled by the particle from t = -p to t = p greater than 5p ? Justify your answer.
Copyright © 2002 by College Entrance Examination Board. All rights reserved. Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.
理论物理和数学物理问题及解答：第二卷 高等水平
理论物理和数学物理问题及解答：第二卷高等水平Willi-Hans Steeb, Rand AfrikaansUniversity, South AfricaProblems Solutions inTheoretical Mathematical PhysicsVol.Ⅱ: Advanced Level Second Edition2022, 362pp.Softcover $ 28.00ISBN 981-238-987-3World Scientific(第2版)本书是理论物理和数学物理问题集，并附解答，其目的是通过精选的问题使学习者加深对理论的理解，掌握重要的解题原则和策略，也是对相关课程的补充。
全书分两卷。
第一卷包括引导性问题，主要针对大学生。
本书是第二卷，包括水平较高、难度较大的问题，面向研究生和年轻科研人员。
书中涉及的问题按内容分为引类，涉及线性代数(如张量积等)，群论和群表示论，李代数、数论、组合(如母函数)、分析^p (如Frechet导数等)，非线性常微分和偏微分方程，稳定性和分支，以及Nambu力学，度量张量场，Killing 矢量场，Fermi及Bose算子等传统内容，还涉及许多新领域，如Lax表示，Bcklund变换，孤立子方程，李代数值微分形式，Hirota技术，Painlevé检验，Bethe拟设，Yang\|Baxtex关系，混沌、分形、复杂性等。
所有问题都被作者使用过，并且每个问题都有详尽解答。
叙述是自封的，可以独立使用。
本书可供数学物理和理论物理专业大学生、研究生和大学教师及研究人员参考。
朱尧辰，研究员(中国科学院应用数学研究所)Zhu Yaochen, Professor(Institute of Applied Mathematics,the Chinese Academy of Sciences)。
等周不等式的微积分证明法
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曲线满足等周不等式。
参考文献：
［１］ ＯＳＳＥＲＭＡＮ Ｒ． Ｔｈｅ ｉｓｏｐｅｒｉｍｅｔｒｉｃ ｉｎｅｑｕａｌｉｔｙ［Ｊ］． Ａｍｅｒ． Ｍａｔｈ． Ｓｏｃ．， １９７８（８４）： １１８２－ １２３９．