B16 – Derivtives of exponential functions：B16–指数函数求导

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Calculus_AB_Syllabus_2

AP ® Calculus AB Syllabus 2Course OutlineBy successfully completing this course, you will be able to:Work with functions represented in a variety of ways and understand the • connections among these representations.Understand the meaning of the derivative in terms of a rate of change • and local linear approximation, and use derivatives to solve a variety of problems.Understand the relationship between the derivative and the definite • integral.Communicate mathematics both orally and in well-written sentences to • explain solutions to problems.Model a written description of a physical situation with a function, a • differential equation, or an integral.Use technology to help solve problems, experiment, interpret results, and • verify conclusions.Determine the reasonableness of solutions, including sign, size, relative • accuracy, and units of measurement.Develop an appreciation of calculus as a coherent body of knowledge and • as a human accomplishment.Technology RequirementI will use a Texas Instruments 84 Plus graphing calculator in class regularly. You will want to have a graphing calculator as well. I recommend the TI-84 and the TI-89. I have a classroom set of TI-84 Plus calculators, and some are available for extended checkout from the media center.We will use the calculator in a variety of ways including:Conduct explorations.• Graph functions within arbitrary windows.• Solve equations numerically. • Analyze and interpret results. • Justify and explain results of graphs and equations.• [C5]C5—The course teaches students how to use graphing calculators to help solve problems, experiment, interpret results, and support conclusions.A Balanced ApproachCurrent mathematical education emphasizes a “Rule of Four.” There are a variety of ways to approach and solve problems. The four branches of the problem-solving tree of mathematics are:Numerical analysis (where data points are known, but not an equation) • Graphical analysis (where a graph is known, but again, not an equation)• Analytic/algebraic analysis (traditional equation and variable • manipulation) Verbal/written methods of representing problems (classic story problems • as well as written justification of one’s thinking in solving a problem—such as on our state assessment) [C3]Below is an outline of topics along with a tentative timeline. Assessments are given at the end of each unit as well as intermittently during each unit. Semester finals are also given.Unit 1: Limits and Continuity (3–4 weeks) [C2]A. Rates of Change 1. Average Speed 2. Instantaneous Speed B. Limits at a Point 1. 1-sided Limits 2. 2-sided Limits 3. Sandwich Theorem**A Graphical Exploration is used to investigate the Sandwich Theorem. Students graph y1 = x^2, y2 = -x^2, y3 = sin (1/x) in radian mode on graphing calculators. The limit as x approaches 0 of each function is explored in an attempt to “see” the limit as x approaches 0 of x^2 * sin (1/x). This helps tie the graphical implications of the Sandwich Theorem to the analytical applications of it. [C3] [C5]C. Limits involving infinity1. Asymptotic behavior (horizontal and vertical)2. End behavior models3. Properties of limits (algebraic analysis)4. Visualizing limits (graphic analysis)D. Continuity1. Continuity at a point2. Continuous functions3. Discontinuous functionsa. Removable discontinuity (0/0 form)C2—The course teachesall topics associatedwith Functions, Graphs, and Limits; Derivatives;Integrals; and Polynomial Approximations andSeries as delineated in the Calculus BCTopic Outline in the AP Calculus CourseDescription.C3—The course provides students with the opportunity to work with functions represented in a variety of ways—graphically, numerically, analytically, and verbally—and emphasizes the connections among these representations.C5—The course teaches students how to use graphing calculators to help solve problems, experiment, interpretresults, and support conclusions.**A tabular investigation of the limit as x approaches 1 of f(x) = (x^2 - 7x - 6)/(x - 1) is conducted in table groups (we have round tables with 3 or 4 students per table). Next, an analytic investigation of the same function is conducted at table groups. Students discuss with their tablemates any conclusions they can draw. Finally, a graphical investigation (using the graphing calculators) is conducted in table groups, and then we discuss, as a class, whether the table group conclusions are verified or contradicted. [C3][C4][C5]b. Jump discontinuity (We look at y = int(x).)c. Infinite discontinuityE. Rates of Change and Tangent Lines1. Average rate of change2. Tangent line to a curve3. Slope of a curve (algebraically and graphically)4. Normal line to a curve (algebraically and graphically)5. Instantaneous rate of changeUnit 2: The Derivative (5–6 weeks) [C2]A. Derivative of a Function1. Definition of the derivative (difference quotient)2. Derivative at a Point3. Relationships between the graphs of f and f’4. Graphing a derivative from data**A CBL experiment is conducted with students tossing a large ball into the air. Students graph the height of the ball versus the time the ball is in the air. The cal-culator is used to find a quadratic equation to model the motion of the ball over time. Average velocities are calculated over different time intervals and students are asked to approximate instantaneous velocity. The tabular data and the regres-sion equation are both used in these calculations. These velocities are graphed versus time on the same graph as the height versus time graph. [C3][C5]5. One-sided derivativesB. Differentiability1. Cases where f’(x) might fail to exist2. Local linearity**An exploration is conducted with the calculator in table groups. Students graph y1 = absolute value of (x) + 1 and y2 = sqrt (x^2 + 0.0001) + 0.99. They investigate the graphs near x = 0 by zooming in repeatedly. The students discuss the local linearity of each graph and whether each function appears to be differentiable at x = 0. [C4][C5]3. Derivatives on the calculator (Numerical derivatives using NDERIV)4. Symmetric difference quotient5. Relationship between differentiability and continuity6. Intermediate Value Theorem for Derivatives C2—The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals; and Polynomial Approximations and Series as delineatedin the Calculus BC Topic Outline in theAP Calculus Course Description.C4—The course teaches studentshow to communicate mathematics and explain solutions to problems both verbally, and in written sentences.C5—The course teaches students how to use graphing calculatorsto help solve problems, experiment, interpret results, and support conclusions.C3—The course provides studentswith the opportunityto work with functions represented in a variety of ways—graphically, numerically, analytically, and verbally—and emphasizes the connections among these representations.C. Rules for Differentiation1. Constant, Power, Sum, Difference, Product, Quotient Rules2. Higher order derivativesD. Applications of the Derivative1. Position, velocity, acceleration, and jerk2. Particle motion3. L’HÔpital’s Rule*Although this topic is not on the AP Calculus AB Exam, I believe this allows students to see the connections between derivatives and limits. Also, it provides a useful way to calculate limits both at a point and as x approaches +/- infinity. I believe this adds to the rigor of the course and the preparedness of students for college-level mathematics courses.4. Economicsa. Marginal costb. Marginal revenuec. Marginal profit*Again, I believe these topics will aid students who choose to matriculate in business in college.E. Derivatives of trigonometric functionsF. Chain RuleG. Implicit Differentiation1. Differential method2. y’ methodH. Derivatives of inverse trigonometric functionsI. Derivatives of Exponential and Logarithmic FunctionsUnit 3: Applications of the Derivative (5–6 weeks) [C2]A. Extreme Values1. Relative Extrema2. Absolute Extrema3. Extreme Value Theorem4. Definition of a critical pointB. Implications of the Derivative1. Rolle’s Theorem2. Mean Value Theorem3. Increasing and decreasing functionsC. Connecting f’ and f’’ with the graph of f(x)1. First derivative test for relative max/min C2—The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals; and Polynomial Approximations and Series as delineatedin the Calculus BC Topic Outline in theAP Calculus Course Description.2. Second derivativea. Concavityb. Inflection pointsc. Second derivative test for relative max/min* A matching game is played with laminated cards that represent functions in four ways: a graph of the function; a graph of the derivative of the function; a written description of the function; and a written description of the derivative of the function. [C3]D. Optimization problemsE. Linearization models1. Local linearization**An exploration using the graphing calculator is conducted in table groups where students graph f(x) = (x^2 + 0.0001)^0.25 + 0.9 around x = 0. Students algebra-ically find the equation of the line tangent to f(x) at x = 0. Students then repeat-edly zoom in on the graph of f(x) at x = 0. Students are then asked to approximate f(0.1) using the tangent line and then calculate f(0.1) using the calculator. This is repeated for the same function, but different x values further and further away from x = 0. Students then individually write about and then discuss with their tablemates the use of the tangent line in approximating the value of the function near (and not so near) x = 0. [C3][C4][C5]2. Tangent line approximation3. DifferentialsF. Related RatesUnit 4: The Definite Integral (3–4 weeks) [C2]A. Approximating areas1. Riemann sumsa. Left sumsb. Right sumsc. Midpoint sumsd. Trapezoidal sums**Here students are asked to input a program that will calculate trapezoidal sums for trapezoids of equal width. They are given this program. They are encouraged to think about altering it to be able to calculate rectangular sums as well. [C5]2. Definite integrals**Students are asked to graph, by hand, a constant function of their choosing. Then they are asked to calculate a definite integral from x = -3 to x = 5 using known geometric methods. Students then share their work with their tablemates and are asked to come up with a table observation. Those observations are shared with other tables and a formula is discovered. [C3]C2—The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals; and Polynomial Approximations and Series as delineatedin the Calculus BC Topic Outline in theAP Calculus Course Description.C4—The course teaches studentshow to communicate mathematics and explain solutions to problems both verbally, and in written sentences.C5—The course teaches students how to use graphing calculatorsto help solve problems, experiment, interpret results, and support conclusions.C3—The course provides studentswith the opportunityto work with functions represented in a variety of ways—graphically, numerically, analytically, and verbally—and emphasizes the connections among these representations.B. Properties of Definite Integrals1. Power rule2. Mean value theorem for definite integrals**An exploration is conducted to show students the geometry of the mean value theorem for definite integrals and how it is connected to the algebra of the theorem. [C3]C. The Fundamental Theorem of Calculus1. Part 12. Part 2Unit 5: Differential Equations and Mathematical Modeling(4 weeks) [C2]A. Slope FieldsB. Antiderivatives1. Indefinite integrals2. Power formulas3. Trigonometric formulas4. Exponential and Logarithmic formulasC. Separable Differential Equations1. Growth and decay2. Slope fields (Resources from the AP Calculus website areliberally used.)3. General differential equations4. Newton’s law of coolingD. Logistic GrowthUnit 6: Applications of Definite Integrals (3 weeks) [C2]A. Integral as net change1. Calculating distance traveled (particle motion)2. Consumption over time3. Net change from dataB. Area between curves1. Area between a curve and an axisa. Integrating with respect to xb. Integrating with respect to y2. Area between intersecting curvesa. Integrating with respect to xb. Integrating with respect to y C2—The course teaches all topics associated with Functions, Graphs, and Limits; Derivatives; Integrals; and Polynomial Approximations and Series as delineatedin the Calculus BC Topic Outline in theAP Calculus Course Description.C3—The course provides studentswith the opportunityto work with functions represented in a variety of ways—graphically, numerically, analytically, and verbally—and emphasizes the connections among these representations.C. Calculating volume1. Cross sections2. Disc method3. Shell methodUnit 7: Review/Test Preparation (time varies, generally 3–5 weeks)A. Multiple-choice practice (Items from past exams—1997, 1998, and 2003are used as well as items from review books I’ve purchased over the years.)1. Test taking strategies are emphasized2. Individual and group practice are both usedB. Free-response practice (Released items from the AP Central website areused liberally.)1. Rubrics are reviewed so students see the need for complete answers2. Students collaborate to formulate team responses3. Individually written responses are crafted. Attention to fullexplanations is emphasized [C4]Unit 8: After the exam…A. Projects designed to incorporate this year’s learning in applied waysB. Research projects on the historical development of mathematics with afocus on calculusC. Advanced integration techniquesD. A look at college math requirements and expectations includingplacement examsTextbook:Finney, Demana, Waits and Kennedy. Calculus—Graphical, Numerical, Algebraic. Third edition. Pearson, Prentice Hall, 2007.This textbook will be our primary resource. You will benefit from reading it. It contains a number of interesting explorations that we will conduct with the goal that you discover fundamental calculus concepts. I will also explain topics in a way that students have found helpful over the years. I encourage cooperative learning, and I believe our entire class benefits from us all working together to help one another construct understanding. [C4]My hope is that you want to learn as much as you can about calculus. Mathematicians have been responsible for many great developments throughout history. Much of our understanding of the universe is a direct result of the contri-butions of mathematicians. Who knows, perhaps we’ll discover something during our course of studies. Whatever happens, I hope you learn to view math as more than just numbers, variables, processes, and algorithms. I hope you learn to apply your mathematical understanding to help you create a better understanding of the mathematical nature of our lives.C4—The course teaches studentshow to communicate mathematics and explain solutions to problems both verbally, and in written sentences.。

Formal Description of OCL Specification Patterns for Behavioral Specification of Software C

Formal Description of OCL Specification Patterns for Behavioral Specification of Software ComponentsJörg AckermannChair of Business Informatics and Systems Engineering,University of Augsburg, Universitätsstr. 16, 86135 Augsburgjoerg.ackermann@wiwi.uni-augsburg.deAbstract. The Object Constraint Language (OCL) is often used for behavioralspecification of software components. One current problem in specifying be-havioral aspects comes from the fact that editing OCL constraints manually istime consuming and error-prone. To simplify constraint definition we proposeto use specification patterns for which OCL constraints can be generated auto-matically. In this paper we outline this solution proposal and develop a wayhow to formally describe such specification patterns on which a library of reus-able OCL specifications is based.Keywords. Software Component Specification, OCL, Specification Patterns1 IntroductionThe Object Constraint Language (OCL) [20] has great relevance for component-based software engineering (CBSE): A crucial prerequisite for applying CBSE successfully is an appropriate and standardized specification of software components [27]. Behav-ioral aspects of components are often specified using OCL (see Sect. 2). From this results one of the current problems in component specifications: Editing OCL con-straints manually is time consuming and error-prone (see Sect. 3).To simplify constraint definition we propose to utilize specification patterns for which OCL constraints can be generated automatically (see Sect. 4). [4] identifies nine patterns that frequently occur in behavioral specifications of software components. In this paper we develop a solution how to formally describe specification patterns that enable a precise pattern specification and aid the implementation of constraint genera-tors (Sect. 5). We conclude with discussion of related work (Sect. 6) and a summary (Sect. 7).The main contributions of this paper are: the proposal to use specification patterns to simplify component specifications and the formal description of specification pat-terns by use of so called OCL pattern functions – together with the identified patterns we obtain a library of reusable OCL specifications.The results are not specific for software components and might therefore be interesting for any user of OCL con-straints.2 Specification of Software ComponentsThe basic paradigm of component-based software engineering is to decouple the pro-duction of components (development for reuse) from the production of complete sys-tems out of components (development by reuse). Applying CBSE promises (amongst others) a shorter time to market, increased adaptability and reduced development costs [8,25].A critical success factor for CBSE is the appropriate and standardized specification of software components: the specification is prerequisite for a composition methodol-ogy and tool support [23] as well as for reuse of components by third parties [26]. With specification of a component we denote the complete, unequivocal and precise description of its external view - that is which services a component provides under which conditions [27].Various authors addressed specifications for specific tasks of the development process as e.g. design and implementation [9,10], component adaptation [28] or com-ponent selection [15]. Approaches towards comprehensive specification of software components are few and include [7,23,27]. Objects to be specified are e.g. business terms, business tasks (domain-related perspective), interface signatures, behavior and coordination constraints (logical perspective) and non-functional attributes (physical perspective).Behavioral specifications (which are topic of this paper) describe how the compo-nent behaves in general and in borderline cases. This is achieved by defining con-straints (invariants, pre- and postconditions) based on the idea of designing applica-tions by contract [18]. OCL is the de-facto standard technique to express such con-straints – cf. e.g. [9,10,23,27].Fig. 1. Interface specification of component SalesOrderProcessingTo illustrate how behavioral aspects of software components are specified we intro-duce a simplified exemplary component SalesOrderProcessing. The business task of the component is to manage sales orders. This component is used as example through-out the rest of the paper.16 J. AckermannFig. 1 shows the interface specification of SalesOrderProcessing using UML [21]. We see that the component offers the interface ISalesOrder with operations to create, check, cancel or retrieve specific sales orders. The data types needed are also defined in Fig. 1. Note that in practice the component could have additional operations and might offer additional order properties. For sake of simplicity we restricted ourselves to the simple form shown in Fig. 1 which will be sufficient as example for this paper. To specify the information objects belonging to the component (on a logical level) one can use a specification data model which is realized as an UML type diagram and is part of the behavioral specification [3]. Fig. 2 displays such a model for the compo-nent SalesOrderProcessing. It shows that the component manages sales orders (with attributes id, date of order, status, customer id) and sales order items (with attributes id, quantity, product id) and that there is a one-to-many relationship between sales orders and sales order items.18 J. Ackermanncan only be called for a sales order that already exists in the component. (More pre-cise: there must exist a sales order which id equals the value of the input parameter orderId. Note that the invariant guarantees that there is at most one such sales order). context SalesOrderinv: SalesOrder.allInstances()->forAll(i1, i2 | i1 <> i2implies i1.id <> i2.id)context ISalesOrder::getOrderData(orderId: string, orderHeader: OrderHeaderData, orderItem: OrderItemData, orderStatus: Order-Status)pre: SalesOrder.allInstances()->exists(id = orderId)Fig. 3. (Partial) Behavioral specification of component SalesOrderProcessing3 Problems in Behavioral Specification of ComponentsMost component specification approaches recommend notations in formal languages since they promise a common understanding of specification results across different developers and companies. The use of formal methods, however, is not undisputed. Some authors argue that the required effort is too high and the intelligibility of the specification results is too low – for a discussion of advantages and liabilities of for-mal methods compare [14].The disadvantages of earlier formal methods are reduced by UML OCL [20]: The notation of OCL has a simple structure and is oriented towards the syntax of object-oriented programming languages. Software developers can therefore handle OCL much easier than earlier formal methods that were based on set theory and predicate logic. This is one reason why OCL is recommended by many authors for the specifica-tion of software components.Despite its advantages OCL can not solve all problems associated with the use of formal methods: One result of two case studies specifying business components [1,2] was the insight that editing OCL constraints manually is nevertheless time consuming and error-prone. Similar experiences were made by other authors that use OCL con-straints in specifications (outside the component area), e.g. [13,17]. They conclude that it takes a considerable effort to master OCL and use it effectively.It should be noted that behavioral aspects (where OCL is used) have a great impor-tance for component specifications: In the specification of a rather simple component in case study [2], for example, the behavioral aspects filled 57 (of altogether 81) pages and required a tremendous amount of work. For component specifications to be prac-tical it is therefore mandatory to simplify the authoring of OCL constraints.Formal Description of OCL Specification Patterns 194 Solution Proposal: Utilizing Specification PatternsSolution strategies to simplify OCL specifications include better tool support (to re-duce errors) and an automation of constraint editing (to reduce effort) – the latter can e.g. be based on use cases or on predefined specification patterns (compare Sect. 6). To use specification patterns seems to be particularly promising for the specifica-tion of business components: When analyzing e.g. the case study [2] one finds that 70% of all OCL constraints in this study can be backtracked to few frequently occur-ring specification patterns. Based on this observation we analyzed a number of com-ponent specifications and literature about component specification and identified nine specification patterns that often occur [4]. These specification patterns are listed in Table 1. Although the nine patterns occurred most often in the investigated material there will be other useful patterns as well and the list might be extended in future.Table 1. Behavioral specification patterns identified in [4]Constraint type Pattern nameInvariant Semantic Key AttributeInvariant Invariant for an Attribute Value of a ClassPrecondition Constraint for a Input Parameter ValuePrecondition Constraint for the Value of an Input Parameter FieldPrecondition Instance of a Class ExistsPrecondition Instance of a Class does not ExistPostcondition Instance of a Class CreatedDefinition Variable Definition for an Instance of a ClassPrecondition Constraint for an Instance Attribute for an Operation CallUnder (OCL) specification pattern we understand an abstraction of OCL constraints that are similar in intention and structure but differ in the UML model elements used. Each pattern has one or more pattern parameters(typed by elements of the UML metamodel) that act as placeholder for the actual model elements. With pattern instan-tiation we denote a specific OCL constraint that results from binding the pattern pa-rameters with actual UML model elements.As an example let us consider the pattern “Semantic Key Attribute”: It represents the situation that an attribute of a class (in the specification data model – cf. Fig. 2) plays the semantic role of a key – that is all instances of the class differ in their value of the key attribute. Pattern parameters are class and attribute and a pattern instantia-tion (for the class SalesOrder and attribute id) can be seen in the upper part of Fig. 3.Table 2. Description scheme for pattern Semantic Key Attribute [4] CharacteristicDescription Pattern nameSemantic Key Attribute Pattern parameterclass: Class; attribute: Property Restrictionsattribute is an attribute of class class Constraint typeInvariant Constraint context classConstraint body name(class).allInstances()->forAll(i1, i2 |i1 <> i2 implies (attribute) <>(attribute)) Based on the ideas of [11] we developed a description scheme that details the proper-ties of a specification pattern: pattern name, pattern parameters, restrictions for pattern use as well as type, context and body of the resulting constraint [4]. Note that the constraint body is a template showing text to be substituted in italic. The description scheme for the pattern Semantic Key Attribute is displayed in Table 2.Fig. 4. Selection screen for generating an OCL constraintThe following points connected with the exemplary pattern are worth mentioning: For sake of simplicity we presented the pattern with only one key attribute. In its regular version the pattern allows that the key is formed by one or more attributes of the class. (Note that this is the reason for not using the operator isUnique which would be rather constructed for more than one attribute.) One can also see that the patterns presented20 J. Ackermannhere are rather static – they allow for substituting UML model elements but do not allow for structural changes. For structural variations on the pattern (e.g.: the attribute id of class SalesOrderItem in Fig. 2 is only unique in the context of a specific instance of class SalesOrder ) one has to define additional patterns. We will now illustrate how such patterns can be exploited for specifications: Sup-pose the person who specifies our exemplary component is in the middle of the speci-fication process and wants to formulate the invariant from Fig. 3. He checks the li-brary of predefined specification patterns (which is part of his specification tool) and finds the pattern for a semantic key attribute (compare section 1 of Fig. 4). After se-lecting this pattern the tool will show him the pattern description and an associated template OCL constraint (showing the pattern parameters in italic). The user has to select model elements for the parameters (in section 3 of Fig. 4) – in our example the class SalesOrder and its attribute id are selected. Note that the tool can be built in such a way that it restricts the input to those model elements that are allowed for a pattern – in section 3 of Fig. 4 for instance you can see that the tool only offers the attributes of class SalesOrder for selection. After providing pattern and parameter values the user can start the generation. The tool checks the input for consistency and then generates the desired OCL constraint (compare section 4 of Fig. 5) which can beincluded into the component specification.Fig. 5. Display of the generated OCL constraintFollowing this approach has the following advantages: For the specification provider maintenance of specifications is simplified because it becomes faster, less error-prone and requires less expert OCL knowledge. For a specification user the understanding of Formal Description of OCL Specification Patterns 2122 J. Ackermannspecifications is simplified because generated constraints are uniform and are there-fore easier recognizable. Moreover, if the patterns were standardized, it would be enough to specify a pattern and the parameter values (without the generated OCL text) which would make recognition even easier.5 Technical Details of the SolutionTo realize the solution outlined in Sect. 4 we need a way to formally describe the specification patterns. Such a formal pattern description is on one hand prerequisite for a tool builder to implement corresponding constraint generators – on the other hand it might also be interesting for a user creating specifications to check if a pattern meets his expectations (although one would not generally expect that a user has the knowledge to understand the formal pattern specifications). In this section we discuss how the specification patterns can be formalized and be described such that their in-tention, structure and application become unambiguous.To do so we first show how such patterns can be formally described and applied (Sect. 5.1). After that we discuss the relationship of the solution to the UML meta-model (Sect. 5.2), argue why we have chosen it compared to other approaches (Sect.5.3) and cover some implementation aspects (Sect. 5.4).5.1 Defining OCL Pattern Functions for Specification PatternsThe basic idea how to formally describe the specification patterns is as follows: For each OCL specification pattern a specific function (called OCL pattern function) is defined. The pattern parameters are the input of the pattern function. Result of the pattern function is a generated OCL constraint which is returned and (if integrated with the specification tool) automatically added to the corresponding UML model element. The OCL pattern functions themselves are specified by OCL – from this specification one can determine the constraint properties (e.g. invariant) and its textual representation. All pattern functions are assigned as operations to a new class OclPat-tern which logically belongs to the layer of the UML metamodel (layer M2 in the four-layer metamodel hierarchy of UML [19] – compare also Sect. 5.2).This approach will now be discussed in detail for the specification pattern “Seman-tic Key Attribute” (see Sect. 4). For this pattern we define the OCL pattern function Create_Inv_SemanticKeyAttribute. Input of the function are a class cl and an attribute attr which is the key attribute of cl – both understood as UML model elements. (To avoid naming conflicts with UML metamodel elements we did not use the pattern parameter names as displayed in the tool in Fig. 4 (like class) but more technical ones (as cl) as input parameters of the pattern functions.) Result is an UML model element of type Constraint. The complete specification of this pattern function is shown in Fig. 6.Formal Description of OCL Specification Patterns 23 context OclPattern::Create_Inv_SemanticKeyAttribute(cl: Class,attr: Property): Constraint(1) pre: attr.class = cl(2) post: result.oclIsNew(3) post: space = result.context(4) post: result.specification.isKindOf(OpaqueExpression)(5) post: nguage = ‘OCL’(6) post: = ’invariant’(7) post: result.context = cl(8) post: = ‘Semantic Key Attribute’(9) post: result.specification.body = OclPattern.Multiconcat(, ‘.allInstances()->forAll( i1, i2 | i1 <> i2implies i1.’, , ‘ <> i2.’, , ‘)’) Fig. 6. Specification of pattern function OclPattern.Create_Inv_SemanticKeyAttributeThe specification of each OCL pattern function consists of three parts: •Preconditions specific for each pattern function (1)•General postconditions (2)-(5)•Postconditions specific for each pattern function (6)-(9).The function specific preconditions describe which restrictions must be fulfilled when calling the pattern function. These preconditions must assure that the actual parame-ters conform to the specification pattern. For instance defines the signature of the pattern function in Fig. 6 only, that cl is any class and attr is any property. The pre-condition (1) demands additionally that attr is an attribute that belongs to class cl.The general postconditions (2)-(5) are identical for all OCL pattern functions and represent in a way the main construction details. These postconditions (together with the functions signature) establish the following:•The return of each pattern function is a UML model element of type Constraint. •This constraint is added to the model (2) and is assigned to the model element which is the context of the constraint (3).•The attribute specification of the constraint is of type OpaqueExpression (4) and is edited in the language OCL (5). (This is in conjunction with the newest version of OCL [20] from June 2005 – earlier there was an inconsistency in the OCL 2.0 specification. Compare Fig. 29 of [20].)In difference to the general postconditions (2)-(5) the postconditions (6)-(9) vary between different pattern functions. The function specific postconditions establish the following:•(6) describes of which constraint type (e.g. invariant, pre- or postcondition) the returned constraint is. The constraint of our example is an invariant.•(7) defines the context of the constraint to be the class cl. The context of an in-variant is always some class and the context of a pre- or postcondition is the clas-sifier to which the operation belongs. Note that OCL imposes additional condi-tions depending on the constraint type. (An invariant, for instance, can only con-strain one model element.) These additional constraints are part of the OCL speci-fication [20, p. 176ff.] and will therefore not be repeated here.24 J. Ackermann•Constraint is a subtype of NamedElement and therefore has an attribute called name [21, p. 94]. This attribute is used in (8) where the constraint is assigned a name which is derived from the specification pattern (in our example the name SemanticKeyAttribute).•The textual OCL representation of a constraint can be found in the attribute body of the property specification(which is of type OpaqueExpression) of the con-straint. Postcondition (9) specifies this textual representation by combining fixed substrings (as ‘ <> i2.’) with the name of model elements which were supplied as pattern parameter values (e.g. ).Note that standard OCL contains the function concatenate which allows concatenating two substrings. In postconditions like (9) of Fig. 6 it is necessary to concatenate many substrings. Technically one could do so by repeated application of OCL concatenate but the resulting expressions were hard to read. Instead we define a help function OclPattern.Multiconcat. Input of this function is a sequence of string arguments and its result is a string which is formed by repeated concatenation of the arguments (in the order given by the sequence).constr := OclPattern.Create_Inv_SemanticKey Attribute(SalesOr-der, id)Fig. 7. Call of pattern function OclPattern.Create_Inv_SemanticKeyAttributeFig. 7 shows how the pattern function Create_Inv_SemanticKeyAttribute is called in our example from Fig. 3: As values for the pattern parameters the class SalesOrder and the property id are used. The precondition is fulfilled because id is indeed an attribute of SalesOrder. The generated constraint constr is an invariant and its textual OCL representation is (as expected) the one shown as result in Fig. 5. (Due to missing UML syntax for operation calls we use in Fig. 7 a syntax that resembles the OCL syntax for operation calls.)Other specification patterns can be described analogously. When defining OCL pat-tern functions one must be careful to select the correct UML metamodel elements for the pattern parameters (classes, properties (of classes), parameters, properties (of parameters) etc.) and to denote all relevant preconditions.One aspect to be mentioned is that some specification patterns require pattern pa-rameters with multiplicity higher than one. (In the regular version of the semantic key pattern there can be one or more attributes that form together the key of the class.) This can be solved by allowing input parameters of a pattern function to have multi-plicity greater than one ([1..*]) and by employing the OCL operator iterate to con-struct the textual OCL specification in something like a loop.5.2 Relationship with the UML MetamodelThe aim of this section is to discuss the relationship of the new class OclPattern with the UML language definition.The UML metamodel is based on a four-layer metamodel hierarchy [19, p. 17ff.]: Layer M0 consist of the run time instances of model elements as e.g. the sales orderwith id ‘1234’. Layer M1 contains the actual user model in which e.g. the class Sale-sOrder is defined. Layer M2 defines the language UML itself and contains e.g. the model element Class. Note that layers M2 and M1 are the meta-layers for layers M1 and M0, respectively. Additionally there exists the layer M3 for the Meta Object Fa-cility (MOF) which is an additional abstraction to define metamodels like UML.For the constraint patterns we defined in Sect. 5.1 a new class OclPattern. To de-cide to which layer this class logically belongs we can analyze input and output of the pattern functions: Input of an OCL pattern function are elements of a UML model (like class SalesOrder or attribute id – on layer M1) that are typed by elements of the UML metamodel (like Class or Property – on layer M2). Analogously the output is always a constraint for a UML model element and is typed by the metamodel element Constraint (on layer M2). Consequently the pattern functions operate on layer M2 and therefore the new class OclPattern logically also belongs to layer M2.On first glance it might seem desirable to integrate the class OclPattern into the UML metamodel (layer M2). The definition of UML, however, does not allow defin-ing new elements in its metamodel. Adding the class OclPattern to layer M2 would effectively mean to define a new modeling language UML’ which consists of UML and one extra class – leaving standard UML yields to many disadvantages (potential compatibility and tool problems) and is not an adequate solution.When looking more closely one finds that it is not necessary to integrate the class OclPattern that tightly into the UML metamodel because it does not change the lan-guage in the sense of introducing new model elements or changing dependencies.As a conclusion it was decided: the class OclPattern will be denoted with the stereotype «oclHelper», operates on layer M2 but stands in parallel to the UML meta-model. The class needs only to be known to the specification tool implementing the constraint generators and is of no direct relevance for model users. The class might be integrated into the UML metamodel at a later time if the UML definition allows it. Note that on a related question OCL users asked to allow user defined OCL functions (Issue 6891 of OCL FTF) which was not realized in OCL 2.0.5.3 Discussion of the SolutionIn this section we will discuss the reasons why the approach presented in Sect. 5.1 was chosen and compare it with other solution approaches that seem (at least at first glance) possible.By defining OCL pattern functions for the specification patterns it became possible to formally describe the patterns completely and quite elegantly: the pattern parame-ters can be found as function parameters and the function specification (which uses again OCL) describes the prerequisites to apply the pattern and the properties of the constraint to be generated. Moreover it is possible to actually specify that the con-straint is added to the UML model element in consideration (assuming the pattern generator is integrated with the specification tool). One big advantage is that this ap-proach only uses known specification techniques and does not require the invention of new ones. There is only one new class OclPattern that encapsulates the definition of all patterns.An alternative approach would be to use a first-hand representation for the abstract constraints before parameter binding – [5] uses this approach and calls this representa-tion constraint schema. The advantage is its explicit representation of the constraint schema. The disadvantage, however, is that constraint schemata are not defined in the UML metamodel – specifying them requires the invention of a special description technique (either outside UML or by introducing a new UML metamodel element). Therefore we decided against using this approach.UML itself offers a mechanism called Templates that allows parameterizing model elements. The following approach seems to be promising and elegant: For each pat-tern one defines a template constraint which is parameterized by the pattern parame-ters – when applying the pattern these parameters are bound to the actual model ele-ments. Unfortunately this solution is technically not possible because UML does not allow parameterizing Constraints (only Classifiers, Packages and Operations) [21, p. 600].To use UML templates nevertheless one might think about parameterizing the con-text of a constraint (which is a classifier or an operation). But this approach is rather constructed and results in many disadvantages: For each invariant pattern used there needs to be a type in the specification data model and all business types using the pattern need to be bound to it. As a result the model would become overcrowded con-tradicting the clarity guideline from the guidelines of modeling [6]. (Similar problems occur with patterns of type pre- or postcondition where template operations need to be added to the interface model.)5.4 Prototype ImplementationConstraint generators for specification patterns were implemented as a prototype (compare Fig. 4 and 5 in Sect. 4). The prototype enables to select a specification pat-tern and values for the pattern specific parameters. As far as possible pattern precondi-tions were considered when providing input for pattern parameters. All other precon-ditions must be checked after value selection. As a result the prototype generates the desired OCL constraint and displays it for the user. Planned for the future is an inte-gration of constraint generators into a component specification tool – that would per-mit to automatically add the generated constraint to the correct model element of the UML model in work.It shall be noted that the pattern parameters to be filled and the preconditions to be checked depend on the specification pattern – in the prototype these were hard coded. One could imagine something like a meta description that enables to (semi)automatically generate the constraint generator. The associated effort, however, seemed not appropriate for only nine specification patterns.6 Related WorkDue to its importance component specifications are discussed by many authors (e.g. [9,10,23,27] – for an overview compare e.g. [23]). Most current specification ap-。

High-dimensional centrally-symmetric polytopes with neighborliness proportional to dimensio

High-Dimensional Centrally-Symmetric PolytopesWith Neighborliness Proportional to DimensionDavid L.DonohoJanuary2005AbstractLet A be a d by n matrix,d<n.Let C be the regular cross polytope(octahedron)in R n.It has recently been shown that properties of the centrosymmetric polytope P=AC areof interest forﬁnding sparse solutions to the underdetermined system of equations y=Ax;[9].In particular,it is valuable to know that P is centrally k-neighborly.We study the face numbers of randomly-projected cross-polytopes in the proportional-dimensional case where d∼δn,where the projector A is chosen uniformly at random fromthe Grassmann manifold of d-dimensional orthoprojectors of R n.We deriveρN(δ)>0withthe property that,for anyρ<ρN(δ),with overwhelming probability for large d,the numberof k-dimensional faces of P=AC is the same as for C,for0≤k≤ρd.This implies thatP is centrally ρd -neighborly,and its skeleton Skel ρd (P)is combinatorially equivalent toSkel ρd (C).We display graphs ofρN.Two weaker notions of neighborliness are also important for understanding sparse so-lutions of linear equations:facial neighborliness and sectional neighborliness[9];we studyboth.The weakest,(k, )-facial neighborliness,asks if the k-faces are all simplicial and ifthe numbers of k-dimensional faces f k(P)≥f k(C)(1− ).We characterize and computethe critical proportionρF(δ)>0at which phase transition occurs in k/d.The other,(k, )-sectional neighborliness,asks whether all,except for a small fraction ,of the k-dimensionalintrinsic sections of P are k-dimensional cross-polytopes.(Intrinsic sections intersect P withk-dimensional subspaces spanned by vertices of P.)We characterize and compute a propor-tionρS(δ)>0guaranteeing this property for k/d∼ρ<ρS(δ).We display graphs ofρS andρF.Key Words and Phrases:Centrosymmetric Polytopes.Neighborly Polytopes.Cross-Polytope.Randomly-Projected Polytope.Grassmann Angle.Internal Angles of Polytopes. External Angles of rge Deviations of Half-Normal random variables.Octahedron. Quotient spaces of 1.Acknowledgements.Partial support from NSF DMS00-77261,and01-40698(FRG), ONR-MURI,and NIH.Thanks to the Institute for Pure and Applied Mathematics(IPAM)and its‘neighborly’hospitality during the program‘Analysis and Geometry in High Dimensions’in the Fall of2004,while this was prepared.Thanks to G.R.Burton and R.Schneider for help with references and Amir Dembo for discussions about large deviations.1Introduction1.1Neighborliness and Central-NeighborlinessIn the classical theory of convex polytopes,the notion of neighborliness oﬀers a beautiful glimpse of the surprises of high dimensions.Neighborliness asks if every k+1vertices of a polytope span a k-face.In low dimensions this is diﬃcult for beginners to arrange–outside the trivial case of the simplex–because it seems that some candidate edges easily get‘swallowed up’crossing ‘inside’the polytope.It can be surprising to students that in higher dimensions d>3this can be managed easily,by simply taking n>d points x i=M(t i)along the moment curve M(t)=(1,t,t2,...,t d−1)[10,12].The convex hull of these points is a polytope with n vertices which are d/2 neighborly,for each n>d;and this is the maximum possible value.See eg. [12,Chapter7]for more.For centrosymmetric polytopes,a modiﬁed notion of neighborliness is needed;one asks if every k+1vertices not including an antipodal pair span a k-face.The slight modiﬁcation detracts a bit from the beauty of the notion;and perhaps also from the interest in studying it.There is no known general construction of centrally k-neighborly for large n and d,and the achievable upper bound is smaller:k≤ (d+1)/3 ,according to McMullen and Shephard[14].For n not much larger than d,Schneider[17]showed the existence of centrally-symmetric polytopes which are k-centrally-neighborly for k≈.2309d;however Schneider’s polytopes have only2d(1+o(1)) vertices.Burton[4]showed that forﬁxed d and large enough n,even2-central-neighborliness is impossible.Not much else seems to have been published.1.2Central Neighborliness and OptimizationIn a companion paper[9],the author shows that central-neighborliness of centrally-symmetric polytopes is important for understanding solvability of certain combinatorial optimization prob-lems by convex relaxation.Speciﬁcally,suppose A is a d-by-n matrix with d<n and we are interested inﬁnding the solution to the underdetermined system y=Ax having fewest nonze-ros.Although this problem is NP-hard in general,the sparsest solution can be often found by solving the convex optimization problem min x 1subject to y=Ax.The conditions on A and y guaranteeing success are:ﬁrst,that a solution with at most k nonzeros exist;and secondly, that the convex polytope P=AC be centrally k-neighborly.Here C denotes the cross-polytope ( 1ball)in R n.The relation to optimization brings new signiﬁcance into the study of neighborliness in the centrosymmetric case.As[9]shows,we can interpret recent results in the study of sparse solutions by 1optimization as constructions of centrosymetric polytopes which are neighborly for reasonably large k.For example,a result of the author[8]relying on Banach space geometry techniques implies that for large d and n,d proportional to n,if we randomly take points x1, ...x n from the uniform distribution on the unit sphere in R d,then the centrosymmetric polytope generated by taking the convex hull of these points and their antipodes is overwhelmingly likely to be k-neighborly,for k∼ρd.Hereρis a positive constant depending on n/d;until now little was known about the possible values forρ.Clearly,we would like to know more about the possible/prevalent ranges of neighborliness.1.3Analysis in High DimensionsIn this paper we adopt the high-dimensional viewpoint,and construct polytopes by projecting from n dimensions down to d dimensions,n large,d proportional to n.The resulting familiesFigure1.1:The lower boundρN(δ)on the neighborliness threshold,computed by methods of this paper.Matlab software available from the author.of high-dimensional centrosymmetric polytopes are proportionally-neighborly,in the sense that for someρ>0and large d,they are typicallyρd-centrally neighborly.Our approach gives quantitative information about the size ofρachievable.We present numerical evidence that k≥.089d when n=2d and n is large.Our analysis considers the ensemble of polytopes P=AC where A is a random projection from R d to R n and C=C n is the standard crosspolytope.We study a functionρN:(0,1]→[0,1],depicted in Figure1.1and deﬁned in detail in later sections.ρN provides a lower bound on the proportional central-neighborliness of the random polytope P.Corollary1.1Let A be a uniform random projection from R n to R d with d= δn .Fix >0.With overwhelming probability for large n,P=AC is centrally k-neighborly with k/d≥ρN(δ)− .1.4Face NumbersIn fact,this article does not much discuss neighborliness per se.Instead,we consider the properties of face numbers of the projected cross-polytope,getting the following result: Theorem1Letρ<ρN(δ)and let A=A d,n be a uniformly-distributed random projection from R n to R d,with d≥δn.ThenP rob{f (AC)=f (C), =0,..., ρd }→1,as n→∞.(1.1) Central k-neighborliness follows from this equality of face numbers;see Section2below.Our proof of Theorem1starts from work of B¨o r¨o czky and Henk[2],who considered face numbers of the randomly projected cross-poytope with dﬁxed and n→∞.We modify the analysis,letting d and n both go to inﬁnity in a proportional way.The approach of[2]depends on the framework for computing Grassmann angles of a polytope due to Aﬀentranger and Schneider [1]and Vershik and Sporyshev[19].This uses exact analytical work in integral geometry of convex sets by McMullen[13](nonlinear sum/angle relations),Gr¨u nbaum[11](Grassmann Angles),and Harold Ruben[15](volumes of spherical simplices).Our approach is to develop formulas for the internal and external angles of cross-polytope faces in the n-proportional-to-d setting,obtaining inequalities of a substantially diﬀerent form than in the d-ﬁxed setting.We use these inequalities to characterize and computeρN(δ).The study of face numbers in the proportional-dimensional case,where d∼δn,was pioneered by Vershik and Sporyshev[19]in the‘projection of simplex’case P=AT n,with T n the regular simplex in R n.Most importantly,Vershik and Sporyshev[19]developed,in addition to the proportional-to-dimension viewpoint,several analytical tools relevant to the proportional-dimensional case,for studying internal and external angles of simplices;these are also used here.1.5Weaker Notions of NeighborlinessVershik and Sporyshev[19]were interested in the question of whether,for k in aﬁxed proportion to n,the face numbers f k(AT n)=f k(T n)(1+o(1))or not.The answer obeyed a threshold phenomenon for k in the vicinity ofρT d,for some implicitly characterizedρT=ρT(d/n,k/n).For comparison to Theorem1,note that the question of approximate equality of face numbers f k(AT n)=f k(T n)(1+o(1))is weaker than the exact equality studied here in Theorem1;it changes at a diﬀerent threshold in k/d.The comparable question in our setting is approximate equality of face numbers f k(AC n)=f k(C n)(1+o(1)).Figure1.2displays thresholds computed based on the following result.Theorem2There is a functionρF(δ),characterised below,with the following property.Let d=d(n)∼δn and let A=A d,n be a uniform random projection from R n to R d.Then for a sequence k=k(n)with k/d∼ρ,ρ<ρF(δ),we havef k(AC)=f k(C)(1+o P(1)).(1.2) This result is sharp in the sense that for sequences with k/d∼ρ>ρF,we do not have the approximate equality(1.2);but we do not prove this here.Thus,we distinguish betweenρF which is really a threshold andρN which is a lower bound on a threshold.As explained in[9],(1.2)can itself be justiﬁed as a weak kind of neighborliness–facial neighborliness–in which the overwhelming majority of(rather than all)k-tuples span(k−1)-faces.This notion of neighborliness is easier to satisfy than orthodox central neighborliness and soρF>ρN.[9]also deﬁnes a notion of sectional neighborliness,intermediate between facial and central neighborliness.In this notion,we take any k vertices not including an antipodal pair and section P by the linear subspace spanned by those vertices.If the overwhelming majority of such sections are k-dimensional cross-polytopes,we say that P is typically sectionally k-neighborly. In Figure1.2we also display a bound on the sectional neighborliness of quotient polytopes, based on the following result.Theorem3There is a functionρS(δ),characterised below,with the following property.Let ρ<ρS(δ)and let A be a uniform random projection from R n to R d,with d≥δn.Then for k∼ρd,we have with overwhelming probability for large d that P=AC is typically sectionally k-neighborly.All three theorems are proved in more or less the same way;we spend the bulk of this article on the proof Theorem1and in aﬁnal section indicate the changes needed to prove Theorems 2-3.Figure1.2depicts substantial numerical diﬀerences in the critical proportionρF and the lower boundsρN andρS.The most striking diﬀerences betweenρF and the other two proportionsFigure1.2:The thresholdρF(δ)for approximate equality ofρd-dimensional face numbers of C and AC(blue),and the lower boundρS(δ)for sectional neighborliness(green).Plot ofρN overlaid in red for comparisonare thatρF crosses the lineρ=1/2nearδ=.701and increases to1asδ→1.The Appendix proves the following.Theorem4limρF(δ)=1.(1.3)δ→1For someδ0∈(0,1),ρF(δ)>1/2,δ0<δ<1.(1.4) For comparison,one can compute thatρN(δ),(1.5).168≈limδ→1andρS(δ).(1.6).352≈limδ→1Such features can be important from the applications viewpoint,where they can be inter-preted as saying that average case behavior is far more favorable than worst-case behavior.See the discussion in[9].2Neighborliness and Face NumbersWeﬁrst justify our claim that face numbers of the quotient polytope alone are enough to determine neighborliness.We alsoﬁx notation concerning convex polytopes;see[12]for more details.In discussing the (closed,convex)polytope P we commonly refer to its vertices v∈vert(P)and k-dimensional faces F∈F k(P).v∈P will be called a vertex of P if there is a linear functionalλv separating v from P\{v},i.e.a value c so thatλv(v)=c andλv(x)<c for x∈P,x=c.We write conv for the convex hull operation;thus P=conv(vert(P)).Vertices are just0-dimensional faces,anda k -dimensional face is a set F for which there exists a separating linear functional λF ,so that λF (x )=c ,x ∈F and λF (x )<c ,x ∈F .Faces are convex polytopes,each one representable as the convex hull of a subset vert(F )⊂vert(P );thus if F is a face,F =conv(vert(F )).A k -dimensional face will be called a k -simplex if it has k +1vertices.Lemma 2.1Let A be an arbitrary linear transformation.Let P =AC have the same face numbers as C ,up to dimension k −1:f (P )=f (C ), =0,1,...,k −1.Then•All the -faces of P are -simplices,for =0,...,k −1.•P is centrally k -neighborly.Proof.We ﬁrst note the very elementary:vert(P )⊂A vert(C ).Indeed,every element of C is a convex combination of its vertices.Every element of P is the image under A of such a convex combination and hence is a convex combination of the signed columns of A .Hence the vertices of P are among the signed columns of A ,andf 0(P )≤f 0(C ).(2.7)Since f 0(P )=f 0(C ),we concludevert(P )=A vert(C ).Thus the vertices of P are made of n antipodal pairs.No antipodal pair can be an edge of P if n >1,because the origin 0serves as the common midpoint of all line segments connecting antipodes.Avoiding such pairs forces f 1(P )≤4 n 2 .Now f 1(C )=4 n 2 .Hence,the hypothesis f 1(P )=f 1(C )implies that F 1(P )contains every possible edge formed from the vertex set which does not connect antipodal vertices.But this means P is centrally-2-neighborly.Consider now a 2-face F ∈F 2(P ).We will show that it is simplicial.We have vert F ⊂vert P .Also,such a 2-face of F cannot contain an antipodal pair of vertices from P .Hence,every pair of vertices of F ,being a non-antipodal pair of vertices in P ,generates an edge in F 1(P ),and hence in F 1(F ).It follows that every 2-face F is 2-neighborly,in the stronger sense of neighborliness appropriate to asymmetric sets –i.e.without any proviso about avoiding antipodes (because there are no antipodal pairs in F to avoid!).We now invoke Theorem 4,Chapter 7of Gr¨u nbaum[12];a d -neighborly d -polytope is a d -simplex .Hence all 2-faces are 2-simplices.Now if all 2-faces are 2-simplices,and no such face can contain an antipodal pair,there are at most 8 n3 such faces.But f 2(C )=8 n 3 .Hence f 2(P )=f 2(C )implies that all allowable combinations of 3vertices generate faces.So P is centrally-3-neighborly.We continue in this way to higher dimensional faces.Each -face F contains no antipodal pairs;by previous steps,all subsets of vertices span faces of P ,and therefore of F ,and so F is -neighborly,and therefore an -simplex.The hypothesis f (P )=f (C )implies that all allowable combinations of +1vertices not containing an antipodal pair generate faces of P ,and so P is +1-centrally-neighborly.We continue through stage k −1,and the lemma is proved.23Random Projections of Cross-PolytopesWe now outline the proof of Theorem1.Key lemmas and inequalities will be justiﬁed in later sections.3.1Angle SumsAs remarked in the introduction,our proof proceeds by reﬁning a line of research in convex integral geometry.Aﬀentranger and Schneider[1](see also Vershik and Sporyshev[19]and B¨o roczky and Henk[2])studied the properties of random projections R=AQ where Q is an n-polytope and R is its d-dimensional orthogonal projection.[1]derived the formulaEf k(R)=f k(Q)−2s≥0F∈F k(Q)G∈F d+1+2s(Q)β(F,G)γ(G,Q);where E denotes the expectation over realizations of the random orthogonal projection,and the sum is over pairs(F,G)where F is a face of G.In this display,β(F,G)is the internal angle at face F of G andγ(G,Q)is the external angle of Q at face G;for deﬁnitions of these terms see eg.Gr¨u nbaum,Chapter14.The slogan underlying the formula is that each face F∈F k(Q)will either‘survive’under projection,so that AF is a k-face of R,or it will get‘swallowed up’inside R.The expected number of faces in R is thus the number of faces in Q minus the expected number faces‘swallowed up’in projection.The chance of a particular face’s getting‘swallowed up’,is exactly the chance that the subspace spanned by columns of A t in R n intersects trivially with the cone of separating linear functionals associated to face F∈Q.The chance that a uniform random subspace hits a cone is precisely the so-called Grassmann angle as deﬁned by Gr¨u nbaum[11].Hence the expected number of faces f k(R)involves a sum of Grassman angles,one for each k-face F of Q,evaluating the probability that AF is a k-face of R.McMullen[13]developed nonlinear angle-sum relations which are used to decompose these Grassmann Angles into the above sums involving internal and external angles.Specializing to the case where Q=C,the n-dimensional Cross-Polytope,we writeEf k(P)=f k(C)−∆(k,d,n)(3.8)with∆(k,d,n)=2s≥0F∈F k(C)G∈F d+1+2s(C)β(F,G)γ(G,C).(3.9)3.2Exact Equality from ExpectationBecause of(2.7)we view(3.8)as showing that on average f k(P)is about the same as f k(C), except for a nonnegative‘discrepancy’∆.We will show that under the stated conditions on k,d, and n,for some >0∆(k,d,n)≤n exp(−n ).(3.10) Now as f k(P)≤f k(C),P rob{f k(P)=f k(C)}≤E(f k(C)−f k(P))=∆(k,d,n).Hence(3.10)implies that with overwhelming probability we get equality of f k(P)with f k(C),as claimed in the theorem.To extend this into the needed simultaneous result-that f (P)=f (C),=0,...,k−1–one deﬁnes events E k={f k(P)=f k(C)}and notes that by Boole’s inequalityP rob(∪k−10E )≤k−1P rob(E k)≤k−1=0∆( ,d,n).The exponential decay of∆(k,d,n)will guarantee that the sum converges to0whenever the k−1-th term does.Hence by establishing(3.10)we getP rob{f (P)=f (C), =0,...,k−1}→1as is to be proved.To establish(3.10),we rewrite(3.9)as∆(k,d,n)=s≥0D swhere,for =d+1+2s,s=0,1,2,...D s=2·F∈F k(C)G∈F d+1+2s(C)β(F,G)γ(G,C).We will show that,forρ<ρN(still to be deﬁned)and for suﬃciently small >0,then for n>n0( ;ρ,δ)n−1log(D s)≤− .This implies(3.10)and hence our main result follows.3.3Decay and Growth ExponentsB¨o r¨o czky and Henk[2]studied exactly the setting P=AC with C the cross-polytope-though for a diﬀerent range of k,d,n(they considered k,dﬁxed and n→∞),and also used a diﬀerent formula for Ef k(P),so they did not directly study the term∆(k,d,n).They did,however,make the following useful observations.•There are2k+1 nk+1k-faces of C.•For >k,there are2 −k n−k−1−k-faces of C containing a given k-face of C.•The faces of C are all simplices,and the internal angleβ(F,G)=β(T k,T ),where T d denotes the standard d-simplex.•The external angleγ(G ,C n)is the same for all -faces of C;it has a closed form integral expression very similar toγ(T ,T n).Thus we can writeD s=2·2 ·nk+1n−k−1−kβ(T k,T )γ(F ,C)=C sβ(T k,T )γ(F ,C), say,with C s the combinatorial prefactor.We plan now to estimate n −1log(D s ),decomposing it into a sum of terms involving log-arithms of the combinatorial prefactor,the internal angle and the external angle.Deﬁne the Shannon entropy:H (p )=p log(1/p )+(1−p )log(1/(1−p ));noting that here the logarithm base is e ,rather than the customary base 2.As did Vershik and Sporyshev [19],we also remark that n −1log n pn→H (p ),p ∈[0,1],n →∞(3.11)so this provides a convenient summary for combinatorial terms.Deﬁning ν= /n ≥δ,we haven −1log(C s )=νlog e (2)+H (ρδ)+H (ν−ρδ1−ρδ)(1−ρδ)+R 1(3.12)with remainder R 1=R 1(s,k,d,n ).Deﬁne then the growth exponent Ψcom (ν;ρ,δ)≡νlog e (2)+H (ρδ)+H (ν−ρδ1−ρδ)(1−ρδ),describing the exponential growth of the combinatorial factors.It is banal to apply (3.11)and see that the remainder R 1in (3.12)is o (1)uniformly in the range k − >(δ−ρ)n ,n >n 0.Section 4.1below deﬁnes a so-called decay exponent Ψext (ν).Section 5shows that γ(F ,C n )decays exponentially at least at the rate Ψext (ν);for each >0,n −1log(γ(F ,C ))≤−Ψext (ν)+ ,uniformly in ≥δn ,n ≥n 0(δ, ).The graph of Ψext is depicted in Figure 4.1.Similarly,Section 4.2below deﬁnes a decay exponent Ψint (ν;ρδ).Section 6below shows that the internal angle β(T k ,T )indeed decays with this exponent;along sequences k ∼ρδn , ∼νn ,n −1log(β(T k ,T ))=−Ψint (ν;ρδ)+R 2,where the remainder R 2≤o (1)uniformly in k − ≥(δ−ρ)n .Hence for any ﬁxed choice of ρ,δ,for >0,and for n ≥n 0(ρ,δ, )we have the inequalityn −1log(D s )≤Ψcom (ν;ρ,δ)−Ψint (ν;ρδ)−Ψext (ν)+3 ,(3.13)valid uniformly in s .3.4Deﬁning ρNDeﬁne now the net exponent Ψnet (ν;ρ,δ)=Ψcom (ν;ρ,δ)−Ψint (ν;ρδ)−Ψext (ν).We can deﬁne at last the mysterious ρN as the threshold where the net exponent changes sign.We will see that the components of Ψnet are all continuous over sets {ρ∈[ρ0,1],δ∈[δ0,1],ν∈[δ,1]},and so Ψnet has the same continuity properties.Deﬁnition 1Let δ∈(0,1].The critical proportion ρN (δ)is the supremum of ρ∈[0,1]obeyingΨnet (ν;ρ,δ)<0,ν∈[δ,1).Continuity of Ψnet shows that if ρ<ρN then,for some >0,Ψnet (ν;ρ,δ)<−4 ,ν∈[δ,1).Combine this with (3.13).Then for all s =0,2,...,(n −d )/2and all n >n 0(δ,ρ, )n −1log(D s )≤− .This implies (3.10)and our main result follows.(a)(b)Figure 4.1:Panel (a):The minimizer x νof ψν,as a function of ν(red)and the asymptotic approximation log(1√πν)(green);Panel (b):The exponent Ψext ,a function of ν.4Properties of ExponentsWe now deﬁne the exponents Ψint and Ψext and discuss properties of ρN .4.1Exponent for External AngleLet G denote the cumulative distribution function of a half-normal HN (0,1/2)random variable,i.e.a random variable X =|Z |where Z ∼N (0,1/2),and G (x )=P rob {X ≤x }.It has density g (x )=2/√πexp(−x 2).Writing this out,G (x )=2√π x 0e −y 2dy ;(4.1)so G is just the classical error function erf .For ν∈(0,1],deﬁne x νas the solution of2xG (x )g (x )=1−νν.(4.2)Since xG is a smooth strictly increasing function ∼0as x →0and ∼x as x →∞,and g (x )is strictly decreasing,the function 2xG (x )/g (x )is one-one on the positive axis,and x νis well-deﬁned,and a smooth,decreasing function of ν.See Figure 4.1for a depiction.This has limiting behavior x ν→0as ν→1and x ν∼ log((1−ν)/(2ν))as ν→0.Deﬁne nowΨext (ν)=−(1−ν)log(G (x ν))+νx 2ν.This is depicted in Figure 4.1.This function is smooth on the interior of (0,1)and concave,with endpoints Ψext (1)=0,Ψext (0)=0.A useful ﬁne point is the asymptoticΨext (ν)∼νlog(1ν)−12νlog(log(1ν))+O (ν),ν→0.(4.3)Figure 4.2:Λ∗(y ),rate function for Half-normal distribution;only the ‘left-half’0<y <µis depicted.The function diverges at 0.4.2Exponent for Internal AngleLet Y be a standard half-normal random variable HN (0,1);this has cumulant generating func-tion Λ(s )=log(E exp(sY )).Very convenient for us is the exact formulaΛ(s )=s 2/2+log(2Φ(s )),where Φis the usual cumulative distribution function of a standard Normal N (0,1).The cu-mulant generating function Λhas a rate function (Fenchel-Legendre dual [6])Λ∗(y )=max ssy −Λ(s ).This is smooth and convex on (0,∞),strictly positive except at µ=EY =2/π.More detailsare provided in Section 6.See Figure 4.2.For γ∈(0,1)let ξγ(y )=1−γγy 2/2+Λ∗(y ).The function ξγ(y )is strictly convex and positive on (0,∞)and has a minimum at a unique y γin the interval (0, 2/π).We deﬁne,for γ=ρδν≤ρ,Ψint (ν;ρδ)=ξγ(y γ)(ν−ρδ)+log(2)(ν−ρδ).This is depicted in Figure 4.3.For ﬁxed ρ,δ,Ψint is continuous in ν≥δ.Most importantly,in Section 6.4below we get the asymptotic formulaξγ(y γ)∼12·log(1−γγ),γ→0.(4.4)Since γ=ρδ/ν≤ρ,(4.4)implies that for given η>0and small ρ,Ψint (ν;ρδ)≥ 12·log(1−ρρ)(1−η)+log(2) (ν−ρδ),ν∈[δ,1].(4.5)Figure4.3:The exponentsΨcom(ν;ρ,δ)(red)andΨint(ν;ρδ)(green),forρ=.095,δ=.5555. For comparison,Ψext is displayed in blue.4.3Combining the ExponentsWe now consider the combined behavior ofΨcom,Ψint andΨext.We think of these as functions ofνwithρ,δas parameters.The combinatorial exponentΨcom is the sum of a linear function in ν,and a scaled,shifted version of the Shannon entropy,which is a symmetric,roughly parabolic shaped function.This is the exponent of a growing function which must be outweighed by the sumΨext+Ψint.It is depicted in Figure4.3.Figure4.4shows bothΨcom andΨext+Ψint withδ=.5555andρ=.095.The desired conditionΨnet<0is the same asΨcom<Ψext+Ψint,and this is distinctly obeyed except near ν=δ,where the two curves are close.We haveρN(δ)≈.095.4.4Properties ofρNThe asymptotic relations(4.5)and(4.3)allow us to see two key facts aboutρN,both proved in the Appendix.Firstly,the concept is nontrivial:Lemma4.1ρN(δ)>0,δ∈(0,1).(4.6) This result was to be expected.Exploiting[9]and[8,5]it could have previously been inferred that,for someρ=ρ(δ)>0such random polytopes are,with overwhelming probability,ρd-neighborly.Eﬀectively,(4.6)shows that the techniques of this paper are at least as strong as those of[8,5].Secondly,one can show that,althoughρN(δ)→0asδ→0,it goes to zero slowly.We prove the following in the appendix.Lemma4.2Forη>0,ρN(δ)≥log(1/δ)−(1+η),δ→0.Again this result could have been anticipated.Exploiting the connection with studies of 1optimization[9],we could have inferred from[7,5],that in the case n∼dγ,γ>1,AC is centrally k-neighborly with k≥cd/log(n).Figure4.4:The exponentsΨcom(ν;ρ,δ)andΨint(ν;ρδ)+Ψext(ν),forρ=.095,δ=.5555.The graph ofΨcom(red)falls below that ofΨint+Ψext(green)and soΨnet<0.5Bounds on the External AngleWe now justify the use ofΨext.Lemma5.1Fixδ, >0.n−1log(γ(F ,C n))≤−Ψext( /n)+ ,(5.1) uniformly in ≥δn,n≥n0(δ, ).We start from an exact identity.B¨o r¨o zcky and Henk[2],building on work of Vershik and Sporyshev[18]and ultimately of H.Ruben[15],give the integral formulaγ(F ,C)=+1π∞e−( +1)x22√πxe−y2dyn− −1dx.We recognize the term in braces as the error function G from(4.1).Setν ,n=( +1)/n.The integral formula can be rewritten asnν ,nπ ∞exp{−nν ,n x2+n(1−ν ,n)log G(x)}dx.(5.2)The appearance of n in the exponent suggests to use Laplace’s method;we deﬁne,forνﬁxed,fν,n(y)=exp{−nψν(y)}·nνπwithψν(y)≡νy2−(1−ν)log G(y).We note thatψνis smooth and convex and(in the appendix)develop expressions for its second and third derivatives.Applying Laplace’s method toψνin the usual way,but taking care about regularity conditions and remainders,gives a result with the uniformity inν,which is crucial for us.Lemma5.2Forν∈(0,1)let xνdenote the minimizer ofψν.Then∞fν,n(x)dx≤exp(−nψν(xν))(1+R n(ν))where,forδ,η>0,supν∈[δ,1−η]R n(ν)=o(1)as n→∞.Of course the minimizer xνmentioned in this lemma is the same xνdeﬁned earlier in(4.2) in terms of the error function,and that the minimum value identiﬁed in this Lemma as driving the exponential rate is the same as our exponentΨext:Ψext(ν)=ψν(xν).(5.3) In fact Lemma5.2easily leads to Lemma5.1.Weﬁrst note thatΨext(ν)→0asν→1.For given >0in the statement of the Lemma,there is(a largest)ν <1withΨext(ν )= .Note thatγ(F ,C)≤1,so that for >ν n,n−1log(γ)≤0<−Ψext(ν)+ ,for n≥1.Consider now ∈[δn,ν n).Taking into account(5.2),we now haveγ(F ,C)=∞0fν,n(x)dx.Applying the uniformity inνgiven in Lemma5.2,we concluden−1log(γ(F ,C))=ψν,n (xν,n)+o(1), ≥δn.Then invoking the identity(5.3)and the uniform continuity ofψνin x and of xνinν∈[δ,1], we getn−1log(γ(F ,C))≤−Ψext( /n)+o(1).Lemma5.1follows.6Bounds on the Internal AngleIn this section we justifyLemma6.1For >0,and n>n0( ,δ,ρ)n−1log(β(T k,T l))≤−Ψint( /n;k/n)+ ,uniformly in ≥δn,k≥ρn,( −k)≥(δ−ρ)n.。

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B16 – Derivatives of exponential functions：B16–指数函数求导

exponentialfunctionstangentlinexe2x2202019calculussantowski15internetlinksmath2413logarithmfunctionsfrompauldawkinsvisualcalculusexponentialfunctionfrompkving谢谢你的阅读知识就是财富丰富你的人生
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2. Find the intervals of increase/decrease for the function f(x) = x2e-x
14
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Calculus I (Math 2413) - Derivatives - Derivatives of Exponential and Logarithm Functions from Paul Dawkins
1. Sketch y = bx Each partner at the table will use a different value for b
2. PREDICT the features of the graph of its derivative by answering the following Q

8
(B) INVESTIGATING THE LIMITS

斯普林格数学研究生教材丛书

《斯普林格数学研究生教材丛书》（Graduate Texts in Mathematics）GTM001《Introduction to Axiomatic Set Theory》Gaisi Takeuti, Wilson M.Zaring GTM002《Measure and Category》John C.Oxtoby（测度和范畴）（2ed.）GTM003《Topological Vector Spaces》H.H.Schaefer, M.P.Wolff（2ed.）GTM004《A Course in Homological Algebra》P.J.Hilton, U.Stammbach（2ed.）（同调代数教程）GTM005《Categories for the Working Mathematician》Saunders Mac Lane（2ed.）GTM006《Projective Planes》Daniel R.Hughes, Fred C.Piper（投射平面）GTM007《A Course in Arithmetic》Jean-Pierre Serre（数论教程）GTM008《Axiomatic set theory》Gaisi Takeuti, Wilson M.Zaring（2ed.）GTM009《Introduction to Lie Algebras and Representation Theory》James E.Humphreys（李代数和表示论导论）GTM010《A Course in Simple-Homotopy Theory》M.M CohenGTM011《Functions of One Complex VariableⅠ》John B.ConwayGTM012《Advanced Mathematical Analysis》Richard BealsGTM013《Rings and Categories of Modules》Frank W.Anderson, Kent R.Fuller（环和模的范畴）（2ed.）GTM014《Stable Mappings and Their Singularities》Martin Golubitsky, Victor Guillemin （稳定映射及其奇点）GTM015《Lectures in Functional Analysis and Operator Theory》Sterling K.Berberian GTM016《The Structure of Fields》David J.Winter（域结构）GTM017《Random Processes》Murray RosenblattGTM018《Measure Theory》Paul R.Halmos（测度论）GTM019《A Hilbert Space Problem Book》Paul R.Halmos（希尔伯特问题集）GTM020《Fibre Bundles》Dale Husemoller（纤维丛）GTM021《Linear Algebraic Groups》James E.Humphreys（线性代数群）GTM022《An Algebraic Introduction to Mathematical Logic》Donald W.Barnes, John M.MackGTM023《Linear Algebra》Werner H.Greub（线性代数）GTM024《Geometric Functional Analysis and Its Applications》Paul R.HolmesGTM025《Real and Abstract Analysis》Edwin Hewitt, Karl StrombergGTM026《Algebraic Theories》Ernest G.ManesGTM027《General Topology》John L.Kelley（一般拓扑学）GTM028《Commutative Algebra》VolumeⅠOscar Zariski, Pierre Samuel（交换代数）GTM029《Commutative Algebra》VolumeⅡOscar Zariski, Pierre Samuel（交换代数）GTM030《Lectures in Abstract AlgebraⅠ.Basic Concepts》Nathan Jacobson（抽象代数讲义Ⅰ基本概念分册）GTM031《Lectures in Abstract AlgebraⅡ.Linear Algabra》Nathan.Jacobson（抽象代数讲义Ⅱ线性代数分册）GTM032《Lectures in Abstract AlgebraⅢ.Theory of Fields and Galois Theory》Nathan.Jacobson（抽象代数讲义Ⅲ域和伽罗瓦理论）GTM033《Differential Topology》Morris W.Hirsch（微分拓扑）GTM034《Principles of Random Walk》Frank Spitzer（2ed.）（随机游动原理）GTM035《Several Complex Variables and Banach Algebras》Herbert Alexander, John Wermer（多复变和Banach代数）GTM036《Linear Topological Spaces》John L.Kelley, Isaac Namioka（线性拓扑空间）GTM037《Mathematical Logic》J.Donald Monk（数理逻辑）GTM038《Several Complex Variables》H.Grauert, K.FritzsheGTM039《An Invitation to C*-Algebras》William Arveson（C*-代数引论）GTM040《Denumerable Markov Chains》John G.Kemeny, urie Snell, Anthony W.KnappGTM041《Modular Functions and Dirichlet Series in Number Theory》Tom M.Apostol （数论中的模函数和Dirichlet序列）GTM042《Linear Representations of Finite Groups》Jean-Pierre Serre（有限群的线性表示）GTM043《Rings of Continuous Functions》Leonard Gillman, Meyer JerisonGTM044《Elementary Algebraic Geometry》Keith KendigGTM045《Probability TheoryⅠ》M.Loève（概率论Ⅰ）（4ed.）GTM046《Probability TheoryⅡ》M.Loève（概率论Ⅱ）（4ed.）GTM047《Geometric Topology in Dimensions 2 and 3》Edwin E.MoiseGTM048《General Relativity for Mathematicians》Rainer.K.Sachs, H.Wu伍鸿熙（为数学家写的广义相对论）GTM049《Linear Geometry》K.W.Gruenberg, A.J.Weir（2ed.）GTM050《Fermat's Last Theorem》Harold M.EdwardsGTM051《A Course in Differential Geometry》Wilhelm Klingenberg（微分几何教程）GTM052《Algebraic Geometry》Robin Hartshorne（代数几何）GTM053《A Course in Mathematical Logic for Mathematicians》Yu.I.Manin（2ed.）GTM054《Combinatorics with Emphasis on the Theory of Graphs》Jack E.Graver, Mark E.WatkinsGTM055《Introduction to Operator TheoryⅠ》Arlen Brown, Carl PearcyGTM056《Algebraic Topology：An Introduction》W.S.MasseyGTM057《Introduction to Knot Theory》Richard.H.Crowell, Ralph.H.FoxGTM058《p-adic Numbers, p-adic Analysis, and Zeta-Functions》Neal Koblitz（p-adic 数、p-adic分析和Z函数）GTM059《Cyclotomic Fields》Serge LangGTM060《Mathematical Methods of Classical Mechanics》V.I.Arnold（经典力学的数学方法）（2ed.）GTM061《Elements of Homotopy Theory》George W.Whitehead（同论论基础）GTM062《Fundamentals of the Theory of Groups》M.I.Kargapolov, Ju.I.Merzljakov GTM063《Modern Graph Theory》Béla BollobásGTM064《Fourier Series：A Modern Introduction》VolumeⅠ（2ed.）R.E.Edwards（傅里叶级数）GTM065《Differential Analysis on Complex Manifolds》Raymond O.Wells, Jr.（3ed.）GTM066《Introduction to Affine Group Schemes》William C.Waterhouse（仿射群概型引论）GTM067《Local Fields》Jean-Pierre Serre（局部域）GTM069《Cyclotomic FieldsⅠandⅡ》Serge LangGTM070《Singular Homology Theory》William S.MasseyGTM071《Riemann Surfaces》Herschel M.Farkas, Irwin Kra（黎曼曲面）GTM072《Classical Topology and Combinatorial Group Theory》John Stillwell（经典拓扑和组合群论）GTM073《Algebra》Thomas W.Hungerford（代数）GTM074《Multiplicative Number Theory》Harold Davenport（乘法数论）（3ed.）GTM075《Basic Theory of Algebraic Groups and Lie Algebras》G.P.HochschildGTM076《Algebraic Geometry：An Introduction to Birational Geometry of Algebraic Varieties》Shigeru IitakaGTM077《Lectures on the Theory of Algebraic Numbers》Erich HeckeGTM078《A Course in Universal Algebra》Stanley Burris, H.P.Sankappanavar（泛代数教程）GTM079《An Introduction to Ergodic Theory》Peter Walters（遍历性理论引论）GTM080《A Course in_the Theory of Groups》Derek J.S.RobinsonGTM081《Lectures on Riemann Surfaces》Otto ForsterGTM082《Differential Forms in Algebraic Topology》Raoul Bott, Loring W.Tu（代数拓扑中的微分形式）GTM083《Introduction to Cyclotomic Fields》Lawrence C.Washington（割圆域引论）GTM084《A Classical Introduction to Modern Number Theory》Kenneth Ireland, Michael Rosen（现代数论经典引论）GTM085《Fourier Series A Modern Introduction》Volume 1（2ed.）R.E.Edwards GTM086《Introduction to Coding Theory》J.H.van Lint（3ed .）GTM087《Cohomology of Groups》Kenneth S.Brown（上同调群）GTM088《Associative Algebras》Richard S.PierceGTM089《Introduction to Algebraic and Abelian Functions》Serge Lang（代数和交换函数引论）GTM090《An Introduction to Convex Polytopes》Ame BrondstedGTM091《The Geometry of Discrete Groups》Alan F.BeardonGTM092《Sequences and Series in BanachSpaces》Joseph DiestelGTM093《Modern Geometry-Methods and Applications》（PartⅠ.The of geometry Surfaces Transformation Groups and Fields）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov （现代几何学方法和应用）GTM094《Foundations of Differentiable Manifolds and Lie Groups》Frank W.Warner（可微流形和李群基础）GTM095《Probability》A.N.Shiryaev（2ed.）GTM096《A Course in Functional Analysis》John B.Conway（泛函分析教程）GTM097《Introduction to Elliptic Curves and Modular Forms》Neal Koblitz（椭圆曲线和模形式引论）GTM098《Representations of Compact Lie Groups》Theodor Breöcker, Tammo tom DieckGTM099《Finite Reflection Groups》L.C.Grove, C.T.Benson（2ed.）GTM100《Harmonic Analysis on Semigroups》Christensen Berg, Jens Peter Reus Christensen, Paul ResselGTM101《Galois Theory》Harold M.Edwards（伽罗瓦理论）GTM102《Lie Groups, Lie Algebras, and Their Representation》V.S.Varadarajan（李群、李代数及其表示）GTM103《Complex Analysis》Serge LangGTM104《Modern Geometry-Methods and Applications》（PartⅡ.Geometry and Topology of Manifolds）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM105《SL₂ (R)》Serge Lang（SL₂ (R)群）GTM106《The Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术理论）GTM107《Applications of Lie Groups to Differential Equations》Peter J.Olver（李群在微分方程中的应用）GTM108《Holomorphic Functions and Integral Representations in Several Complex Variables》R.Michael RangeGTM109《Univalent Functions and Teichmueller Spaces》Lehto OlliGTM110《Algebraic Number Theory》Serge Lang（代数数论）GTM111《Elliptic Curves》Dale Husemoeller（椭圆曲线）GTM112《Elliptic Functions》Serge Lang（椭圆函数）GTM113《Brownian Motion and Stochastic Calculus》Ioannis Karatzas, Steven E.Shreve （布朗运动和随机计算）GTM114《A Course in Number Theory and Cryptography》Neal Koblitz（数论和密码学教程）GTM115《Differential Geometry：Manifolds, Curves, and Surfaces》M.Berger, B.Gostiaux GTM116《Measure and Integral》Volume1 John L.Kelley, T.P.SrinivasanGTM117《Algebraic Groups and Class Fields》Jean-Pierre Serre（代数群和类域）GTM118《Analysis Now》Gert K.Pedersen（现代分析）GTM119《An introduction to Algebraic Topology》Jossph J.Rotman（代数拓扑导论）GTM120《Weakly Differentiable Functions》William P.Ziemer（弱可微函数）GTM121《Cyclotomic Fields》Serge LangGTM122《Theory of Complex Functions》Reinhold RemmertGTM123《Numbers》H.-D.Ebbinghaus, H.Hermes, F.Hirzebruch, M.Koecher, K.Mainzer, J.Neukirch, A.Prestel, R.Remmert（2ed.）GTM124《Modern Geometry-Methods and Applications》（PartⅢ.Introduction to Homology Theory）B.A.Dubrovin, A.T.Fomenko, S.P.Novikov（现代几何学方法和应用）GTM125《Complex Variables：An introduction》Garlos A.Berenstein, Roger Gay GTM126《Linear Algebraic Groups》Armand Borel（线性代数群）GTM127《A Basic Course in Algebraic Topology》William S.Massey（代数拓扑基础教程）GTM128《Partial Differential Equations》Jeffrey RauchGTM129《Representation Theory：A First Course》William Fulton, Joe HarrisGTM130《Tensor Geometry》C.T.J.Dodson, T.Poston（张量几何）GTM131《A First Course in Noncommutative Rings》m（非交换环初级教程）GTM132《Iteration of Rational Functions：Complex Analytic Dynamical Systems》AlanF.Beardon（有理函数的迭代：复解析动力系统）GTM133《Algebraic Geometry：A First Course》Joe Harris（代数几何）GTM134《Coding and Information Theory》Steven RomanGTM135《Advanced Linear Algebra》Steven RomanGTM136《Algebra：An Approach via Module Theory》William A.Adkins, Steven H.WeintraubGTM137《Harmonic Function Theory》Sheldon Axler, Paul Bourdon, Wade Ramey（调和函数理论）GTM138《A Course in Computational Algebraic Number Theory》Henri Cohen（计算代数数论教程）GTM139《Topology and Geometry》Glen E.BredonGTM140《Optima and Equilibria：An Introduction to Nonlinear Analysis》Jean-Pierre AubinGTM141《A Computational Approach to Commutative Algebra》Gröbner Bases, Thomas Becker, Volker Weispfenning, Heinz KredelGTM142《Real and Functional Analysis》Serge Lang（3ed.）GTM143《Measure Theory》J.L.DoobGTM144《Noncommutative Algebra》Benson Farb, R.Keith DennisGTM145《Homology Theory：An Introduction to Algebraic Topology》James W.Vick（同调论：代数拓扑简介）GTM146《Computability：A Mathematical Sketchbook》Douglas S.BridgesGTM147《Algebraic K-Theory and Its Applications》Jonathan Rosenberg（代数K理论及其应用）GTM148《An Introduction to the Theory of Groups》Joseph J.Rotman（群论入门）GTM149《Foundations of Hyperbolic Manifolds》John G.Ratcliffe（双曲流形基础）GTM150《Commutative Algebra with a view toward Algebraic Geometry》David EisenbudGTM151《Advanced Topics in the Arithmetic of Elliptic Curves》Joseph H.Silverman（椭圆曲线的算术高级选题）GTM152《Lectures on Polytopes》Günter M.ZieglerGTM153《Algebraic Topology：A First Course》William Fulton（代数拓扑）GTM154《An introduction to Analysis》Arlen Brown, Carl PearcyGTM155《Quantum Groups》Christian Kassel（量子群）GTM156《Classical Descriptive Set Theory》Alexander S.KechrisGTM157《Integration and Probability》Paul MalliavinGTM158《Field theory》Steven Roman（2ed.）GTM159《Functions of One Complex Variable VolⅡ》John B.ConwayGTM160《Differential and Riemannian Manifolds》Serge Lang（微分流形和黎曼流形）GTM161《Polynomials and Polynomial Inequalities》Peter Borwein, Tamás Erdélyi（多项式和多项式不等式）GTM162《Groups and Representations》J.L.Alperin, Rowen B.Bell（群及其表示）GTM163《Permutation Groups》John D.Dixon, Brian Mortime rGTM164《Additive Number Theory：The Classical Bases》Melvyn B.NathansonGTM165《Additive Number Theory：Inverse Problems and the Geometry of Sumsets》Melvyn B.NathansonGTM166《Differential Geometry：Cartan's Generalization of Klein's Erlangen Program》R.W.SharpeGTM167《Field and Galois Theory》Patrick MorandiGTM168《Combinatorial Convexity and Algebraic Geometry》Günter Ewald（组合凸面体和代数几何）GTM169《Matrix Analysis》Rajendra BhatiaGTM170《Sheaf Theory》Glen E.Bredon（2ed.）GTM171《Riemannian Geometry》Peter Petersen（黎曼几何）GTM172《Classical Topics in Complex Function Theory》Reinhold RemmertGTM173《Graph Theory》Reinhard Diestel（图论）（3ed.）GTM174《Foundations of Real and Abstract Analysis》Douglas S.Bridges（实分析和抽象分析基础）GTM175《An Introduction to Knot Theory》W.B.Raymond LickorishGTM176《Riemannian Manifolds：An Introduction to Curvature》John M.LeeGTM177《Analytic Number Theory》Donald J.Newman（解析数论）GTM178《Nonsmooth Analysis and Control Theory》F.H.clarke, Yu.S.Ledyaev, R.J.Stern, P.R.Wolenski（非光滑分析和控制论）GTM179《Banach Algebra Techniques in Operator Theory》Ronald G.Douglas（2ed.）GTM180《A Course on Borel Sets》S.M.Srivastava（Borel 集教程）GTM181《Numerical Analysis》Rainer KressGTM182《Ordinary Differential Equations》Wolfgang WalterGTM183《An introduction to Banach Spaces》Robert E.MegginsonGTM184《Modern Graph Theory》Béla Bollobás（现代图论）GTM185《Using Algebraic Geomety》David A.Cox, John Little, Donal O’Shea（应用代数几何）GTM186《Fourier Analysis on Number Fields》Dinakar Ramakrishnan, Robert J.Valenza GTM187《Moduli of Curves》Joe Harris, Ian Morrison（曲线模）GTM188《Lectures on the Hyperreals：An Introduction to Nonstandard Analysis》Robert GoldblattGTM189《Lectures on Modules and Rings》m（模和环讲义）GTM190《Problems in Algebraic Number Theory》M.Ram Murty, Jody Esmonde（代数数论中的问题）GTM191《Fundamentals of Differential Geometry》Serge Lang（微分几何基础）GTM192《Elements of Functional Analysis》Francis Hirsch, Gilles LacombeGTM193《Advanced Topics in Computational Number Theory》Henri CohenGTM194《One-Parameter Semigroups for Linear Evolution Equations》Klaus-Jochen Engel, Rainer Nagel（线性发展方程的单参数半群）GTM195《Elementary Methods in Number Theory》Melvyn B.Nathanson（数论中的基本方法）GTM196《Basic Homological Algebra》M.Scott OsborneGTM197《The Geometry of Schemes》David Eisenbud, Joe HarrisGTM198《A Course in p-adic Analysis》Alain M.RobertGTM199《Theory of Bergman Spaces》Hakan Hedenmalm, Boris Korenblum, Kehe Zhu（Bergman空间理论）GTM200《An Introduction to Riemann-Finsler Geometry》D.Bao, S.-S.Chern, Z.Shen GTM201《Diophantine Geometry An Introduction》Marc Hindry, Joseph H.Silverman GTM202《Introduction to Topological Manifolds》John M.LeeGTM203《The Symmetric Group》Bruce E.SaganGTM204《Galois Theory》Jean-Pierre EscofierGTM205《Rational Homotopy Theory》Yves Félix, Stephen Halperin, Jean-Claude Thomas（有理同伦论）GTM206《Problems in Analytic Number Theory》M.Ram MurtyGTM207《Algebraic Graph Theory》Chris Godsil, Gordon Royle（代数图论）GTM208《Analysis for Applied Mathematics》Ward CheneyGTM209《A Short Course on Spectral Theory》William Arveson（谱理论简明教程）GTM210《Number Theory in Function Fields》Michael RosenGTM211《Algebra》Serge Lang（代数）GTM212《Lectures on Discrete Geometry》Jiri Matousek（离散几何讲义）GTM213《From Holomorphic Functions to Complex Manifolds》Klaus Fritzsche, Hans Grauert（从正则函数到复流形）GTM214《Partial Differential Equations》Jüergen Jost（偏微分方程）GTM215《Algebraic Functions and Projective Curves》David M.Goldschmidt（代数函数和投影曲线）GTM216《Matrices：Theory and Applications》Denis Serre（矩阵：理论及应用）GTM217《Model Theory An Introduction》David Marker（模型论引论）GTM218《Introduction to Smooth Manifolds》John M.Lee（光滑流形引论）GTM219《The Arithmetic of Hyperbolic 3-Manifolds》Colin Maclachlan, Alan W.Reid GTM220《Smooth Manifolds and Observables》Jet Nestruev（光滑流形和直观）GTM221《Convex Polytopes》Branko GrüenbaumGTM222《Lie Groups, Lie Algebras, and Representations》Brian C.Hall（李群、李代数和表示）GTM223《Fourier Analysis and its Applications》Anders Vretblad（傅立叶分析及其应用）GTM224《Metric Structures in Differential Geometry》Gerard Walschap（微分几何中的度量结构）GTM225《Lie Groups》Daniel Bump（李群）GTM226《Spaces of Holomorphic Functions in the Unit Ball》Kehe Zhu（单位球内的全纯函数空间）GTM227《Combinatorial Commutative Algebra》Ezra Miller, Bernd Sturmfels（组合交换代数）GTM228《A First Course in Modular Forms》Fred Diamond, Jerry Shurman（模形式初级教程）GTM229《The Geometry of Syzygies》David Eisenbud（合冲几何）GTM230《An Introduction to Markov Processes》Daniel W.Stroock（马尔可夫过程引论）GTM231《Combinatorics of Coxeter Groups》Anders Bjröner, Francesco Brenti（Coxeter 群的组合学）GTM232《An Introduction to Number Theory》Graham Everest, Thomas Ward（数论入门）GTM233《Topics in Banach Space Theory》Fenando Albiac, Nigel J.Kalton（Banach空间理论选题）GTM234《Analysis and Probability：Wavelets, Signals, Fractals》Palle E.T.Jorgensen（分析与概率）GTM235《Compact Lie Groups》Mark R.Sepanski（紧致李群）GTM236《Bounded Analytic Functions》John B.Garnett（有界解析函数）GTM237《An Introduction to Operators on the Hardy-Hilbert Space》Rubén A.Martínez-Avendano, Peter Rosenthal（哈代-希尔伯特空间算子引论）GTM238《A Course in Enumeration》Martin Aigner（枚举教程）GTM239《Number Theory：VolumeⅠTools and Diophantine Equations》Henri Cohen GTM240《Number Theory：VolumeⅡAnalytic and Modern Tools》Henri Cohen GTM241《The Arithmetic of Dynamical Systems》Joseph H.SilvermanGTM242《Abstract Algebra》Pierre Antoine Grillet（抽象代数）GTM243《Topological Methods in Group Theory》Ross GeogheganGTM244《Graph Theory》J.A.Bondy, U.S.R.MurtyGTM245《Complex Analysis：In the Spirit of Lipman Bers》Jane P.Gilman, Irwin Kra, Rubi E.RodriguezGTM246《A Course in Commutative Banach Algebras》Eberhard KaniuthGTM247《Braid Groups》Christian Kassel, Vladimir TuraevGTM248《Buildings Theory and Applications》Peter Abramenko, Kenneth S.Brown GTM249《Classical Fourier Analysis》Loukas Grafakos（经典傅里叶分析）GTM250《Modern Fourier Analysis》Loukas Grafakos（现代傅里叶分析）GTM251《The Finite Simple Groups》Robert A.WilsonGTM252《Distributions and Operators》Gerd GrubbGTM253《Elementary Functional Analysis》Barbara D.MacCluerGTM254《Algebraic Function Fields and Codes》Henning StichtenothGTM255《Symmetry Representations and Invariants》Roe Goodman, Nolan R.Wallach GTM256《A Course in Commutative Algebra》Kemper GregorGTM257《Deformation Theory》Robin HartshorneGTM258《Foundation of Optimization》Osman GülerGTM259《Ergodic Theory：with a view towards Number Theory》Manfred Einsiedler, Thomas WardGTM260《Monomial Ideals》Jurgen Herzog, Takayuki HibiGTM261《Probability and Stochastics》Erhan CinlarGTM262《Essentials of Integration Theory for Analysis》Daniel W.StroockGTM263《Analysis on Fock Spaces》Kehe ZhuGTM264《Functional Analysis, Calculus of Variations and Optimal Control》Francis ClarkeGTM265《Unbounded Self-adjoint Operatorson Hilbert Space》Konrad Schmüdgen GTM266《Calculus Without Derivatives》Jean-Paul PenotGTM267《Quantum Theory for Mathematicians》Brian C.HallGTM268《Geometric Analysis of the Bergman Kernel and Metric》Steven G.Krantz GTM269《Locally Convex Spaces》M.Scott Osborne。

纹理物体缺陷的视觉检测算法研究--优秀毕业论文

摘要
在竞争激烈的工业自动化生产过程中，机器视觉对产品质量的把关起着举足轻重的作用，机器视觉在缺陷检测技术方面的应用也逐渐普遍起来。与常规的检测技术相比，自动化的视觉检测系统更加经济、快捷、高效与安全。纹理物体在工业生产中广泛存在，像用于半导体装配和封装底板和发光二极管，现代化电子系统中的印制电路板，以及纺织行业中的布匹和织物等都可认为是含有纹理特征的物体。本论文主要致力于纹理物体的缺陷检测技术研究，为纹理物体的自动化检测提供高效而可靠的检测算法。纹理是描述图像内容的重要特征，纹理分析也已经被成功的应用与纹理分割和纹理分类当中。本研究提出了一种基于纹理分析技术和参考比较方式的缺陷检测算法。这种算法能容忍物体变形引起的图像配准误差，对纹理的影响也具有鲁棒性。本算法旨在为检测出的缺陷区域提供丰富而重要的物理意义，如缺陷区域的大小、形状、亮度对比度及空间分布等。同时，在参考图像可行的情况下，本算法可用于同质纹理物体和非同质纹理物体的检测，对非纹理物体的检测也可取得不错的效果。在整个检测过程中，我们采用了可调控金字塔的纹理分析和重构技术。与传统的小波纹理分析技术不同，我们在小波域中加入处理物体变形和纹理影响的容忍度控制算法，来实现容忍物体变形和对纹理影响鲁棒的目的。最后可调控金字塔的重构保证了缺陷区域物理意义恢复的准确性。实验阶段，我们检测了一系列具有实际应用价值的图像。实验结果表明本文提出的纹理物体缺陷检测算法具有高效性和易于实现性。关键字: 缺陷检测；纹理；物体变形；可调控金字塔；重构
Keywords: defect detection, texture, object distortion, steerable pyramid, reconstruction
II

A Theory for Multiresolution Signal Decomposition The Wavelet ...

I
Manuscript received July 30. 1987: revised December 23. 1988. This work was supported under the following Contracts and Grants: NSF grant IODCR-84 1077 1. Air Force Grant AFOSR F49620-85-K-0018. Army DAAG-29-84-K-0061. NSF-CERiDC82-19196 Ao2. and DARPAiONR ARPA N0014-85-K-0807. The author is with the Department of Computer Science Courant Institute of Mathematical Sciences. New York University. New York, NY 10012. IEEE Log Number 8928052.
674
IEEE TRANSACTIONS PATTERN ON ANALYSISAND MACHINE INTELLIGENCE.VOL. II, NO. 7. JULY 1%')
A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
STEPHANE Gsolution representations are very effective for analyzing the information content of images. We study the properties of the operator which approximates a signal at a given resolution. We show that the difference of information between the approximation of a signal at the resolutions 2’ + ’ and 2jcan be extracted by decomposing this signal on a wavelet orthonormal basis of L*(R”). In LL(R ), a wavelet orthonormal basis is a family of functions ( @ w (2’ ~ n)) ,,,“jEZt, which is built by dilating and translating a unique function t+r(xl. This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror lilters. For images, the wavelet representation differentiates several spatial orientations. We study the application of this representation to data compression in image coding, texture discrimination and fractal analysis. Index Terms-Coding, fractals, multiresolution pyramids, ture mirror filters, texture discrimination, wavelet transform. quadra-

Two-dimensional Quantum Field Theory, examples and applications

Abstract The main principles of two-dimensional quantum ﬁeld theories, in particular two-dimensional QCD and gravity are reviewed. We study non-perturbative aspects of these theories which make them particularly valuable for testing ideas of four-dimensional quantum ﬁeld theory. The dynamics of conﬁnement and theta vacuum are explained by using the non-perturbative methods developed in two dimensions. We describe in detail how the eﬀective action of string theory in non-critical dimensions can be represented by Liouville gravity. By comparing the helicity amplitudes in four-dimensional QCD to those of integrable self-dual Yang-Mills theory, we extract a four dimensional version of two dimensional integrability.
2 48 49 52 54 56
5 Four-dimensional analogies and consequences 6 Conclusions and Final Remarks

Monotone method for periodic boundary value problems of Caputo fractional differential equations

$Monotone method for periodic boundary value problems of Caputo fractional differential equations$

1. INTRODUCTION The derivative of an arbitrary order or fractional derivative has been introduced almost 300 years ago with a query posed by L’Hospital to Leibnitz. The fractional calculus was reasonably developed by 19th century. It was realized, only in the past few decades that these derivatives are better models to study physical phenomenon in transient state. This gave a fresh lease to this ﬁeld and there is a growing interest to study the theory of fractional diﬀerential equations.[1, 3, 4, 5, 6, 8, 9, 10]. The monotone iterative technique [7] is an eﬀective and ﬂexible mechanism that oﬀers theoretical, as well as constructive results in a closed set, namely, the sector. The generalized monotone iterative technique is a generalization and a reﬁnement of the monotone method. In this paper, the PBVP for Caputo fractional diﬀerential equation is considered and the generalized monotone iterative technique is developed to cater to the situation where the function on the righthand side is split into two functions- a function that can be made into a non-decreasing function and a non-increasing function.

电力系统潮流计算软件设计外文原文及中文翻译

电力系统潮流计算软件设计外文原文及中文翻译外文原文及中文翻译Modelling and Analysis of Electric Power SystemsPower Flow Analysis Fault AnalysisPower Systems Dynamics and StabilityPrefaceIn the lectures three main topics are covered,i.e.Power flow an analysisFault current calculationsPower systems dynamics and stabilityIn Part I of these notes the two first items are covered,while Part II givesAn introduction to dynamics and stability in power systems. In appendices brief overviews of phase-shifting transformers and power system protections are given.The notes start with a derivation and discussion of the models of the most common power system components to be used in the power flow analysis.A derivation of the power ?ow equations based on physical considerations is then given.The resulting non-linear equations are for realistic power systems of very large dimension and they have to be solved numerically.The most commonly used techniques for solving these equations are reviewed.The role of power flow analysis in power system planning,operation,and analysis is discussed.The next topic covered in these lecture notes is fault current calculations in power systems.A systematic approach to calculate fault currents in meshed,large power systems will be derived.The needed models will be given and the assumptions made when formulating these models discussed.It will be demonstrated thatalgebraic models can be used to calculate the dimensioning fault currents in a power system,and the mathematical analysis has similarities with the power ?ow analysis,soitis natural to put these two items in Part I of the notes.In Part II the dynamic behaviour of the power system during and after disturbances(faults) will be studied.The concept of power system stability isde?ned,and different types of pow er system in stabilities are discussed.While the phenomena in Part I could be studied by algebraic equations,the description of the power system dynamics requires models based on differential equations.These lecture notes provide only a basic introduction to the topics above.To facilitate for readers who want to get a deeper knowledge of and insight into these problems,bibliographies are given in the text.Part IStatic Analysis1 IntroductionThis chapter gives a motivation why an algebraic model can be used to de scribe the power system in steady state.It is also motivated why an algebraic approach can be used to calculate fault currents in a power system.A power system is predominantly in steady state operation or in a state that could with sufficient accuracy be regarded as steady state.In a power system there are always small load changes,switching actions,and other transients occurring so that in a strict mathematical sense most of the variables are varying with thetime.However,these variations are most of the time so small that an algebraic,i.e.not time varying model of the power systemis justified.A short circuit in a power system is clearly not a steady state condition.Such an event can start a variety of different dynamic phenomena in the system,and to study these dynamic models are needed.However,when it comes to calculate the fault current sin the system,steady state(static) model swith appropriate parameter values can be used.A fault current consists of two components,a transient part,and a steady state part,but since the transient part can be estimated from the steady state one,fault current analysis is commonly restricted to the calculation of the steady state fault currents.1.1 Power Flow AnalysisIt is of utmost importance to be able to calculate the voltages and currents that different parts of the power system are exposed to.This is essential not only in order to design the different power system components such asgenerators,lines,transformers,shunt elements,etc.so that these can withstand the stresses they are exposed to during steady state operation without any risk of damages.Furthermore,for an economical operation of the system the losses should be kept at a low value taking various constraint into account,and the risk that the system enters into unstable modes of operation must be supervised.In order to do this in a satisfactory way the state of the system,i.e.all(complex) voltages of all nodes in the system,must be known.With these known,all currents,and hence all active and reactive power flows can be calculated,and other relevant quantities can be calculated in the system.Generally the power ?ow,or load ?ow,problem is formulated as a nonlinear set of equationsf (x, u, p)=0(1.1)wheref is an n-dimensional(non-linear)functionx is an n-dimensional vector containing the state variables,or states,ascomponents.These are the unknown voltage magnitudes and voltage angles of nodes in the systemu is a vector with(known) control outputs,e.g.voltages at generators with voltage controlp is a vector with the parameters of the network components,e.g.line reactances and resistancesThe power flow problem consists in formulating the equations f in eq.(1.1) and then solving these with respect to x.This will be the subject dealt with in the first part of these lectures.A necessary condition for eq.(1.1) to have a physically meaningful solution is that f and x have the same dimension,i.e.that we have the same number of unknowns as equations.But in the general case there is no unique solution,and there are also cases when no solution exists.If the states x are known,all other system quantities of interest can be calculated from these and the known quantities,i.e. u and p.System quantities of interest are active and reactive power flows through lines and transformers,reactive power generation from synchronous machines,active and reactive power consumption by voltage dependent loads, etc.As mentioned above,the functions f are non-linear,which makes the equations harder to solve.For the solution of the equations,the linearizationy X Xf ?= (1.2)is quite often used and solved.These equations give also very useful information about the system.The Jacobian matrix Xf ?? whose elements are given by j iij X f X f ??=??）（(1.3)can be used form any useful computations,and it is an important indicator of the system conditions.This will also be elaborate on.1.2 Fault Current AnalysisIn the lectures Elektrische Energiesysteme it was studied how to calculate fault currents,e.g.short circuit currents,for simple systems.This analysis will now be extended to deal with realistic systems including several generators,lines,loads,and other system components.Generators(synchronous machines) are important system components when calculating fault currents and their model will be elaborated on and discussed.1.3 LiteratureThe material presented in these lectures constitutes only an introduction to thesubject.Further studies can be recommended in the following text books:1. Power Systems Analysis,second edition,by Artur R.Bergen and VijayVittal.(Prentice Hall Inc.,2000,ISBN0-13-691990-1,619pages)2. Computational Methods for Large Sparse Power Systems,An object oriented approach,by S.A.Soma,S.A.Khaparde,Shubba Pandit(Kluwer Academic Publishers, 2002, ISBN0-7923-7591-2, 333pages)2 Net work ModelsIn this chapter models of the most common net work elements suitable for power flow analysis are derived.These models will be used in the subsequent chapters when formulating the power flow problem.All analysis in the engineering sciences starts with the formulation of appropriate models.A model,and in power system analysis we almost invariably then mean a mathematical model,is a set of equations or relations,which appropriately describes the interactions between different quantities in the time frame studied and with the desired accuracy of a physical or engineered component or system.Hence,depending on the purpose of the analysis different models of the same physical system or components might be valid.It is recalled that the general model of a transmission line was given by the telegraph equation,which is a partial differential equation, and by assuming stationary sinusoidal conditions the long line equations, ordinary differential equations,were obtained.By solving these equations and restricting the interest to the conditions at the ends of the lines,the lumped-circuit line models (π-models) were obtained,which is an algebraic model.This gives us three different models each valid for different purposes.In principle,the complete telegraph equations could be used when studying the steady state conditions at the network nodes.The solution would then include the initial switching transients along the lines,and the steady state solution would then be the solution after the transients have decayed. However, such a solution would contain a lot more information than wanted and,furthermore,it would require a lot of computational effort.An algebraic formulation with the lumped-circuit line model would give the same result with a much simpler model ata lower computational cost.In the above example it is quite obvious which model is the appropriate one,but in many engineering studies these lection of the“correct”model is often the most difficult part of the study.It is good engineering practice to use as simple models as possible, but of course not too simple.If too complicated models are used, the analysis and computations would be unnecessarily cumbersome.Furthermore,generally more complicated models need more parameters for their definition,and to get reliable values of these requires often extensive work.i i+diu+du C ’dx G ’dxR ’dx L ’dx u dxFigure2.1. Equivalent circuit of a line element of length dx In the subsequent sections algebraic models of the most common power system components suitable for power flow calculations will be derived.If not explicitly stated,symmetrical three-phase conditions are assumed in the following.2.1 Lines and CablesThe equ ivalent π-model of a transmission line section was derived in the lectures Elektrische Energie System, 35-505.The general distributed model is characterized by the series parametersR′=series resistance/km per phase(?/km)X′=series reactance/km per phase(?/km)and the shunt parametersB′=shunt susceptance/km per phase(siemens/km)G′=shunt conductance/km per phase(siemens/km )As depicted in Figure2.1.The parameters above are specific for the line or cable configuration and are dependent onconductors and geometrical arrangements.From the circuit in Figure2.1the telegraph equation is derived,and from this the lumped-circuit line model for symmetrical steady state conditions,Figure2.2.This model is frequently referred to as the π-model,and it is characterized by the parameters）（Ω=+=impedance series jX R km km km Z ）（siemens admittance shuntjB G Y sh km sh km sh km =+= I mk Z km y sh km y sh mkI kmkmFigure2.2. Lumped-circuit model(π-model)of a transmission line betweennodes k and m.Note. In the following most analysis will be made in the p.u.system.Forimpedances and admittances,capital letters indicate that the quantity is expressed in ohms or siemens,and lower case letters that they are expressed in p.u.Note.In these lecture notes complex quantities are not explicitly marked asunder lined.This means that instead of writing km Z we will write km Z when this quantity is complex. However,it should be clear from the context if a quantity is real or complex.Furthermore,we will not always use specific type settings for vectors.Quite often vectors will be denoted by bold face type setting,but not always.It should also be clear from the context if a quantity is a vector or a scalar.When formulating the net work equations the nodeadmittance matrix will be used and the series admittance of the line model is neededkm km 1-km km jb g z y +== (2.1)With22km r g km km kmx r +=(2.2)and 22km x -b km km kmx r += (2.3)For actual transmission lines the series reactance km x and the series resistance km r are both positive,and consequently km g is positive and km b is negative.The shunt susceptance sh y km and the shunt conductance sh g km are both positive for real line sections.In many cases the value of sh g km is so small that it could be neglected.The complex currents km I and mk I in Figure2.2 can be expressed as functions of the complex voltages at the branch terminal nodes k and m:k sh km m k km km E y E E y I +-=)( (2.4)m k m mk )(E y E E y I sh km km +-=(2.5)Where the complex voltages arek j k k e θU E = (2.6)k j k k e θU E =(2.7) This can also be written in matrix form as））（（）（m k sh km km km km sh km km mk km E E y y y -y -y y I I ++=(2.8) As seen the matrix on the right hand side of eq.(2.8)is symmetric and thediagonal elements are equal.This reflects that the lines andcables are symmetrical elements.2.2 TransformersWe will start with a simplified model of a transformer where we neglect the magnetizing current and the no-load losses .In this case the transformer can be modelled by an ideal transformer with turns ratio km t in series with a series impedance km z which represents resistive(load-dependent)losses and the leakage reactance,see Figure2.3.Depending on if km t is real ornon-real(complex)the transformer is in-phase or phase-shifting.p k mU m ej θm I km I mkU kej θk U p e j θp Z km 1：t km p k mU m ej θm I km I mkU kej θk U p e j θp Z km t km ：1Figure2.3. Transformer model with complex ratio kmj km km e a t ?=（km -j 1-km km e a t ?=） mp k U m ej θm I km I mk U kej θk U p e j θp Z km a km ：1Figure2.4. In-phase transformer model 2.2.1In-Phase TransformersFigure2.4shows an in-phase transformer model indicating the voltage at the internal –non-physical –node p.In this model the ideal voltage magnitude ratio(turns ratio)iskm k p(2.9) Since θk = θp ,this is also the ratio between the complex voltages at nodes k and p, km j k j p k pa e U e U E E k p ==θθ(2.10)There are no power losses(neither active nor reactive)in the idealtransformer(the k-p part of the model),which yields0I E I E *mk p *km k =+(2.11) Then applying eqs.(2.9)and(2.10)giveskm mk km mk km -a I I -I I ==(2.12)A B Ck m I mk I kmFigure2.5. Equivalent π-model for in-phase transformerwhich means that the complex currents km I and mk I are out of phase by 180since km a ∈ R.Figure2.5 represents the equivalent π-model for thein-phase transformer in Figure2.4.Parameters A, B,and C of this model can be obtained by identifying the coefficients of the expressions for the complex currents km I and mk I associated with the models of Figures2.4 and 2.5.Figure2.4 givesm km km k km 2km p m km km km E y a E y a E -E y -a I ）（）（）（+==(2.13)m km k km km p m km mk E y E y a -E -E y I ）（）（）（+== (2.14)or in matrix form ））（（）（m k km km km km km km2km mk km E E y y a -y a -y a I I =As seen the matrix on the right hand side of eq.(2.15) is symmetric,but thediagonal elements are not equal when 1a 2km ≠.Figure2.5 provides now the following:m k km E A -E A -I ）（）（+=(2.16)m k mk E C A E A -I ）（）（++=(2.17)or in matrix form））（（）（m k mk km E E C A A -A -B A I I ++= (2.18)Identifying the matrix elements from the matrices in eqs.(2.15) and (2.18) yieldskm km y a A = (2.19)km km km y 1-a a B ）（= (2.20)km km )y a -(1C =(2.21) 2.2.2 Phase-Shifting TransformersPhase-shifting transformers,such as the one represented in Figure2.6,are used to control active power flows;the control variable is the phase angle and the controlled quantity can be,among other possibilities,the active power flow in the branch where the shifter is placed.In Appendix A the physical design of phase-shifting transformer is described. A phase-shifting transformer affects both the phase and magnitude of the complex voltages k E and p E ,without changing their ratio,i.e., km j km km k p e a t E E ?== (2.22)Thus, km k p ?θθ+=and k km p U a U =,using eqs. (2.11) and (2.22)km j -km *km mkkm e -a -t I I ?==I km m U m ej θm I mk pkU k ej θk Z km 1：a kme j φkmkm k p ?θθ+=k km p U a U = Figure2.6. Phase-shifting transformer with km j km km e a t ?=As with in-phase transformers,the complex currentskm I and mk I can be expressed in terms of complex voltages at the phase-shifting transformer terminals:m km *km k km 2km p m km *km km E y t -E y a E -E y -t I ）（）（）（+== (2.24)m km k km km p m km mk E y E y t -E -E y I ）（）（）（+==(2.25)Or in matrix form））（（）（m k km km km km *km km 2km mk km E E y y t -y t -y a I I =(2.26) As seen this matrix is not symmetric if km t is non-real,and the diagonal matrixelements are not equal if 1a 2km ≠.There is no way to determine parameters A, B,and Cof the equivalent π-model from these equations,since the coefficient km *km y t - ofEm in eq.(2.24)differs from km km y t -in eq.(2.25),as long as there is non zero phase shift,i.e. km t ?R.A phase-shifting transformer can thus not be represented by a π-model.2.2.3Unified Branch ModelThe expressions for the complex currents km I and mk I for both transformersand shifters derived above depend on the side where the tap is located;i.e., they are not symmetrical.It is how ever possible to develop unified complex expressions which can be used for lines,transformers,and phase-shifters, regardless of the side on which the tap is located(or even in the case when there are taps on both sides of thedevice).Consider initially the model in Figure2.8 in which shunt elements have beentemporarily ignored and km j km km e a t ?= and m k j mk mk e a t ?=。

112 Derivatives of exponential and Logarithmic Functions

Example 4
Find g’(x) for A) g ( x) x10 10x
g ' ( x) 10x9 10x ln(10)
B)
g ( x) log2 x 6 log5 x
1 1 1 1 g ' ( x) 6 x ln 2 x ln 5 1 1 6 g ' ( x) x ln 2 ln 5
1 x
Example 3
Find y’ for A) y 10x3 100ln x 100 1 2 2 y' 30x 100 30x x x
B) y ln x5 e x ln e2
y 5 ln x e x ln e2
1 x y ' 5 e 0 x
Example 5
An Internet store sells blankets. If the price-demand equation is p = 200(0.998)x, find the rate of change of price with respect to demand when the demand is 400 blankets and explain the result. p’ = 200 (.998)x ln(0.998) p’(400) = 200 (.998)400 ln(0.998) = -0.18. When the demand is 400 blankets, the price is decreasing about 18 cents per blanket
Review
y logb x
Domain: (0, ∞) Range: (-∞, ∞)

First-principles study of the structural, vibrational, phonon and thermodynamic

1. Introduction Ultra-high temperature ceramics (UHTCs) with melting temperatures in excess of 3000 K are usually composed by the refractory borides, carbides and nitrides of early transition metals [1–7]. Among the UHTCs, transition metal carbides (TMC) such as TiC, ZrC and HfC are metallic compounds with unique physical and chemical properties including an extremely high melting point and hardness, chemical stability, corrosion resistance combined with metallic electrical and thermal conductivities [5–10]. These features give transition metal carbides the capability to withstand high temperatures in oxidizing environments, making them candidates for applications in the atmosphere of extreme thermal and chemical environments [6,7]. The structural, vibrational, phonon and thermodynamic properties of IVb group transition metal carbides have been investigated experimentally [10–17] and theoretically [13,18–28] in the earlier reports. In the 1970s, the phonon dispersion relations of TiC, ZrC and HfC were measured using inelastic neutron scattering by Pintschovius et al. [10] and Smith et al. [15–17]. Lattice dynamics calculation and the phonon dispersion relations of transition metal carbides such as ZrC and HfC were reported using a phenomenological ‘‘double-shell’’ model theory [18] where long-range interatomic interactions were taken into account in order to get a

常微分方程中的英文单词和短语

算子多项式
operator polynomial
rse transform
第七章
Chapter7Power Series Solutions of Differential Equations
柯西定理
Cauchy theorem
解析解
analytic solution
常数矩阵
constant matrix
非齐次线性微分方程组
non-homogeneous linear differential equations
常数列向量
constant arrange vector
常数变异公式
formula of variation of constant
计算公式
computing formula
若尔当标准型
Jordan normal form
矩阵函数
matrix function
实值解
real-valued function
复值解
complex-value function
代数余子式
algebraic cofactor
特征方程
characteristic equation
算子式
operator method
一阶线性方程
first-order linear equation
齐次
homogeneous
非齐次
non-homogeneous
积分因子法
method of integration factor
常数变易法
method of constants variation
初等变换法
elementary method of transformation

Stretched--exponential relaxation in arrays of coupled rotators

Abstract We consider the non–equilibrium dynamics of a chain of classical rotators coupled at its edges to an external reservoir at zero temperature. We ﬁnd that the energy is released in a strongly discontinuous fashion, with sudden jumps alternated with long stretches during which dissipation is extremely weak. The jumps mark the disappearance of strongly localized structures, akin to the rotobreather solutions of the Hamiltonian model, which act as insulating boundaries of a hot central core. As a result of this complex kinetics, the ensemble–averaged energy follows a stretched exponential law until a residual pseudo–stationary state is attained, where the hot core has reduced to a single localized object. We give a statistical description of the relaxation pathway and connect it to the properties of return periods of rare events in correlated time series. This approach sheds some light into the microscopic mechanism underlying the slow dynamics of the system. Finally, we show that the stretched exponential law remains unaltered in the presence of isotopic disorder. Key words: Stretched–exponential relaxation, sine–lattice, rotobreathers, isotopic disorder PACS: 05.45.-a, 63.20.Ry

外文翻译-分数阶导数

浙江师范大学本科毕业设计(论文)外文翻译的分子和分母去替换，则得到结果如下的一般表达式。

我们通过伽玛函数，用任意数。

假设我们可以连续次地取分原文：A Child’s Garden of Fractional DerivativesMarcia Kleinz and Thomas J. OslerThe College Mathematics Journal , March 2000, Volume 31, Number 2, pp. 82–88 Marcia Kleinz is an instructor of mathematics at Rowan University. Marcia is married and has two children aged four and eight. She would rather research the fractional calculus than clean, and preparing lectures is preferable to doing laundry. Her hobbies include reading, music, and physical fitness.Tom Osler (osler@) is a professor of mathematics at Rowan University. He received his Ph.D. from the Courant Institute at New York University in 1970 and is the author of twenty-three mathematical papers. In addition to teaching university mathematics for the past thirty-eight years, Tom has a passion for long distance running. Included in his over 1600 races are wins in threenational championships in the late sixties at distances from 25 kilometers to 50 miles. He is the author of two running books.Introduction We are all familiar with the idea of derivatives. The usual notation()df x dxor 1()D f x ，22()d f x dx or 2()D f x is easily understood. We are also familiar with properties like[()()]()()D f x f y Df x Df y +=+ But what would be the meaning of notation like 1/21/2()d f x dxor 1/2()D f x ？Most readers will not have encountered a derivative of “order ” before, because almost none of the familiar textbooksmention it. Yet the notion was discussed briefly as early as the eighteenth century by Leibnitz. Other giants of the past including L’Hospital, Euler, Lagrange, Laplace, Riemann, Fourier, Liouville, and others at least toyed with the idea. Today a vast literature exists on this subject called the “fractional cal culus.” Two text books on the subject at the graduate level have appeared recently, [9] and [11]. Also, two collections of papers delivered at conferences are found in [7] and [14]. A set of very readable seminar notes has been prepared by Wheeler [15], but these have not beenpublished.It is the purpose of this paper to introduce the fractional calculus in a gentle manner. Rather than the usual definition —lemma —theorem approach, we explore the idea of a fractional derivative by first looking at examples of familiar n th order derivatives like D n ax n ax e a e = and then replacing the natural number n by other numbers like In this way, like detectives, we will try to see whatmathematical structure might be hidden in the idea. We will avoid a formal definition of the fractional derivative until we have first explored the possibility of various approaches to the notion. (For a quick look at formal definitions see the excellent expository paper by Miller [8].)As the exploration continues, we will at times ask the reader to ponder certain questions. The answers to these questions are found in the last section of this paper. So just what is a fractional derivative?Let us see. . . .Fractional derivatives of exponential functionsWe will begin by examining the derivatives of the exponential function axe because the patterns they develop lend themselves to easy exploration. We are familiar with the expressions for the derivativesof ax e .12233,,ax ax ax ax ax ax D e ae D e a e D e a e ===, and, in general, n ax n ax D e a e = when n is aninteger. Could we replace n by 1/2 and write 1/21/2ax ax D e a e =Why not try? Why not go further andlet n be an irrational number likeor a complex number like1+i ?We will be bold and write ax ax D e a e αα=， (1)for any value of α, integer, rational, irrational, or complex. It is interesting to consider the meaningof (1) when is αa negative integer. We naturally want1(())ax ax e D D e -=.Since 1(())ax ax e D e a =,we have1()ax ax D e e dx -=⎰.Similarly, 2()ax ax D e e dxdx -=⎰⎰,so is it reasonable to interpret D αwhen αis a negative integer –n as the n th iterated integral.D αrepresents a derivative if αis a positive real number and an integral if αis a negative real number.Notice that we have not yet given a definition for a fractional derivative of a general function. But if that definition is found, we would expect our relation (1) to follow from it for the exponential function. We note that Liouville used this approach to fractional differentiation in his papers [5] and [6]. QuestionsQ1 In this case does 12121212()a x a x a x a xD c e c e c De c De α+=+? Q2 In this case does ax ax D D eD e αβαβ+=? Q3 Is 1()ax ax D e e dx -=⎰, and is 2()ax ax D e e dxdx -=⎰⎰ ,(as listed above) really true, or is there something missing?Q4 What general class of functions could be differentiated fractionally be means ofthe idea contained in (1)?Trigonometric functions: sine and cosine.We are familiar with the derivatives of the sine function:012sin sin ,sin cos ,sin sin ,D x x D x x D x x ===-This presents no obvious pattern from which to find 1/2sin D x . However, graphing the functions discloses a pattern. Each time we differentiate, the graph of sin x is shifted /2π to the left. Thus differentiating sin x n times results in the graph of sin x being shifted /2n π to the left and so sin sin()2n n D x x π=+. As before, we will replace the positive integer n with an arbitrary α. So, we now have an expression for the general derivative of the sine function, and we can deal similarly with the cosine:sin sin(),cos cos().22D x x D x x αααπαπ=+=+ (2)After finding (2), it is natural to ask if these guesses are consistent with the results of the previous section for the exponential. For this purpose we can use Euler’s expression,cos sin ix e x i x =+ Using (1) we can calculate(/2)cos()sin()22ix ix i ix D e i e e e x i x ααπααπαπ===+++which agrees with (2).QuestionQ5 What is sin()D ax α?Derivatives of pxWe now look at derivatives of powers of x . Starting with p x we have: 012,,(1),,(1)(2)(1).(3)p p p p p p n p p n D x x D x px D x p p x D x p p p p n x -===-=---+Multiplying the numerator and denominator of (3) by (p-n )! results in (1)(2)(1)()(1)1!(4)()(1)1()!p p n p np p p p n p n p n p x x x p n p n p n -----+---==----This is a general expression of n p D x .To replace the positive integer n by the arbitrary number αwe may use the gamma function. The gamma function gives meaning to p ! and (p-n )! in (4) when p and n are not natural numbers. The gamma function was introduced by Euler in the 18th century togeneralize the notion of z ! to non-integer values of z . Its definition is 10()d t z z e tt ∞--Γ=⎰,and it hasthe property that (+1)!z z Γ=. We can rewrite (4) as(1),(1)n p p n p D x x p n -Γ+=Γ-+ which makes sense if n is not an integer, so we put(1)(5)(1)p p p D x x p ααα-Γ+=Γ-+for any α. With (5) we can extend the idea of a fractional derivative to a large number of functions. Given any function that can be expanded in a Taylor series in powers of x ,0(),n n n f x a x ∞==∑ assuming we can differentiate term by term we get00(1)().(6)(1)nn n n n n n D f x a D x a x n αααα∞∞-==Γ+==Γ-+∑∑The final expression presents itself as a possible candidate for the definition of the fractionalderivative for the wide variety of functions that can be expanded in a Taylor’s series in powers of x . However, we will soon see that it leads to contradictions.QuestionQ6 Is there a meaning for ()D f x αin geometric terms?A mysterious contradictionWe wrote the fractional derivative of as (7)x xD e e α= Let us now compare this with (6) to see if they agree. From the Taylor Series, 01,!x n n e x n ∞==∑(6) gives 0.(8)(1)nx n x D e n αα∞==Γ-+∑But (7) and (8) do not match unless is a whole number! When is a whole number, the right side of (8)will be the series of x e with different indexing. But when αis not a whole number, we have twoentirely different functions. We have discovered acontradiction that historically has caused great problems. It appears as though ourexpression (1) for the fractional derivative of the exponential is inconsistent with ourformula (6) for the fractional derivative of a power.This inconsistency is one reason the fractional calculus is not found in elementary texts. In thetraditional calculus, where α is a whole number, the derivative of an elementary function is anelementary function. Unfortunately, in the fractional calculus this is not true. The fractional derivative of an elementary function is usually a higher transcendental function. For a table of fractional derivatives see [3].At this point you may be asking what is going on? The mystery will be solved in later sections. Stay tuned . . . .Iterated integralsWe have been talking about repeated derivatives. Integrals can also be repeated. We could write 1()()D f x f x dx -=⎰,but the right-hand side is indefinite. We will instead write10()()x D f x f t dt -=⎰.The second integral will then be 2211200()()x t D f x f t dt dt -=⎰⎰.The region of integration is the triangle in Figure 1. If we interchange the order of integration, the right-hand diagram in Figure 1 shows that121210()()xx t D f x f t dt dt -=⎰⎰Since 1()f t is not a function of 2t , it can be moved outside the inner integral so,1212111100()()()()x x xt D f x f t dt dt f t x t dt -==-⎰⎰⎰ or20()()()xD f x f t x t dt -=-⎰ Using the same procedure we can show that32430011()()(),()()(),223x x D f x f t x t dt D f x f t x t dt --=-=-⋅⎰⎰ and, in general, 101()()().(1)!x n n D f x f t x t dt n --=--⎰ Now, as we have previously done, let us replace the –n with arbitrary αand the factorial with the gamma function to get101()().(9)()()x f t dt D f x x t ααα+=Γ--⎰This is a general expression (using an integral) for fractional derivatives that has the potential of being used as a definition. But there is a problem. If 1,α>- the integer is improper. This occurs because as ,0.t x x t →-→The integral diverges for every 0α≥.When 10,α-<< the improperintegral converges, so if α is negative there is no problem. Since (9) converges only for negative it is truly a fractional integral. Before we leave this section we want to mention that the choice of zero for the lower limit was arbitrary. The lower limit could just as easily have been b . However, the resulting expression will be different. Because of this, many people who work in this field use thenotation ()b x D f x α indicating limits of integration going from b to x . Thus we have from (9)11()().(10)()()x b x b f t dt D f x x t ααα+=Γ--⎰QuestionQ7 What lower limit of fractional differentiation b will give us the result(1)()()(1)pp b x p D x c x c p ααα-Γ+-=-Γ-+? The mystery solvedNow you may begin to see what went wrong before. We are not surprised that fractional integrals involve limits, because integrals involve limits. Since ordinary derivatives do not involve limits of integration, no one expects fractional derivatives to involve such limits. We think of derivatives as local properties of functions. The fractional derivative symbol D αincorporates both derivatives (positive α) and integrals (negative α). Integrals are between limits. It turns out that fractional derivatives are between limits also. The reason for the contradiction is that two different limits of integration were being used. Now we can resolve the mystery.What is the secret? Let’s stop and think. What are the limits that will work for the exponential from (1)? Remember we want to write 11.(11)x ax ax ax b x b D e e dx e a-==⎰ What value of b will give this answer? Since the integral in (11) is really 11.x ax ax ab b e dx e e a a =-⎰we will get the form we want when 10ab e a=.It will be zero when .ab =-∞So, if a is positive, then b =-∞.This type of integral with a lower limit of -∞ is sometimes called the Weyl fractional derivative. In the notation from (10) we can write (1) as.ax ax x D e a e αα-∞=Now, what limits will work for the derivative of px in (5)? We have 111.11p p x p pb x b x b D x x dx p p ++-==-++⎰ Again we want 101p b p +=+。

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Calculus - Santowski

9
(B) INVESTIGATING THE LIMITS
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Investigate lim h0 (4h – 1)/h numerically with a table of values
Calculus - Santowski

(2) Develop an algebraic derivation of the derivative of an exponential function (3) Apply the various differentiation rules to differentiate exponential functions
Is
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EXPONENTIAL FUNCTIONS SUMMARY

The derivative of an exponential function was
d x ah 1 x a a lim h 0 dx h

Which we will now rewrite as

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EXPLORATION – PART 4
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Now we will use algebra to PROVE that our observations were correct. So we go back to our limit definition of a derivative: Our definition is:
x y -0.00010 1.38620 -0.00007 1.38623 -0.00003 1.38626 0.00000 undefined 0.00003 1.38633 0.00007 1.38636 0.00010 1.38639 And we see the value of 1.386 as an approximation of the limit
3

(A) EXPLORATION – PART 1
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You are now pursuing derivatives of other functions (not just power functions) To begin our study of derivatives of exponential functions, let’s explore a bit first 1. Sketch y = bx Each partner at the table will use a different value for b 2. PREDICT the features of the graph of its derivative by answering the following Q (a) Identify the intervals of increase and decrease (b) identify the critical values (c) From this information (and knowing what each means about the derivative), prepare a hand drawn graph of the derivative

d x a ax ln a dx

And we will see one special derivative when the exponential base is e, then the derivative becomes:
d x e ex ln e e x 1 e x dx

Calculus - Santowski
(3) Differentiate
ex y 2 x 2
f ( x) 2
x

(4) Differentiate

(5) Differentiate (6) Differentiate

ex y 1 e 2x
h( x) x 3 x5
Calculus - Santowski

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(E) EXAMPLES – DIFFERENTIATION OF EXPONENTIAL FUNCTIONS
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(1) Find the derivative of y = e3x (2) Find the derivative of y = 3x2e2x
Calculus - Santowski
2.
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(G) INTERNET LINKS
Calculus I (Math 2413) - Derivatives - Derivatives of Exponential and Logarithm Functions from Paul Dawkins Visual Calculus - Derivative of Exponential Function From pkving
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Calculus - Santowski
B.1.6 – DERIVATIVES OF EXPONENTIAL FUNCTIONS
1
CALCULUS - SANTOWSKI
FAST FIVE
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ex – e2x Factor e2x – ex x x Factor e e Simplify (2x)(22) Solve xe3x – 3xex = 0

5
EXPLORATION – PART 3
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Now let’s use graphing technology:
Calculus - Santowski
Use the TI-89 to directly and if the derivative of y = bx. What is the derivative of y = bx? Confirm that your equation for the derivative is correct (and show/explain how you confirmed this.)
Factor
2
Calculus - Santowski

Evaluate
the limit
eh 1 lim h 0 h
2
LESSON OBJECTIVES
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(1) Investigate the derivative of exponential functions using a variety of methods
x
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(F) EXAMPLES - APPLICATIONS
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1.
Find the equation of the tangent line to the curve y = 1 + xe2x at x = 0 Find the intervals of increase/decrease for the function f(x) = x2e-x

Investigate lim h0 (eh – 1)/h numerically with a table of values
x y -0.00010 0.99995 -0.00007 0.99997 -0.00003 0.99998 0.00000 undefined 0.00003 1.00002 0.00007 1.00003 0.00010 1.00005
Calculus - Santowski
(1) Use GDC to draw the tangent lines at various x values (2) Record the slopes of the tangent lines on a table. (3) Prepare a scatter plot from the table of values. (4) Describe scatter plot
Calculus - Santowski

f x h f ( x ) f x lim h 0 h

So work with it ……

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DERIVATIVE OF EXPONENTIAL FUNCTIONS
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Calculus - Santowski
8
(B) INVESTIGATING THE LIMITS
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Investigate lim h0 (2h – 1)/h numerically with a table of values
x y -0.00010 0.69312 -0.00007 0.69313 -0.00003 0.69314 0.00000 undefined 0.00003 0.69316 0.00007 0.69316 0.00010 0.69317 And we see the value of 0.693 as an approximation of the limit

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Calculus - Santowski
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Calculus - Santowski
Calculus - Santowski