pptx
... Higher-order functions The “magic”: How do we use the “right environment” for lexical scope when functions may return other functions, store them in data structures, etc.? Lack of magic: The interpreter uses a closure data structure (with two parts) to keep the environment it will need to use later ...
... Higher-order functions The “magic”: How do we use the “right environment” for lexical scope when functions may return other functions, store them in data structures, etc.? Lack of magic: The interpreter uses a closure data structure (with two parts) to keep the environment it will need to use later ...
Algorithms Lecture 5 Name:___________________________
... In "divide-and-conquer" algorithms (e.g., recursive fibonacci, binary search, merge sort, ...): ...
... In "divide-and-conquer" algorithms (e.g., recursive fibonacci, binary search, merge sort, ...): ...
Unit 3: Functions - Connecticut Core Standards
... Students are introduced to the concept of a function in the first investigation of this unit. After identifying relationships that are or are not functions, they learn how to define the domain and range of a function. Investigation Two provides practice applying the concept of a function through var ...
... Students are introduced to the concept of a function in the first investigation of this unit. After identifying relationships that are or are not functions, they learn how to define the domain and range of a function. Investigation Two provides practice applying the concept of a function through var ...
3-5 Continuity and End Behavior
... a. The graph of same-day shipping function is discontinuous at each integral value of w in its domain because the function does not approach the same value from the left and the right. For example, as w approaches 70 from the left, C(w) approaches 229.25, but as w approaches 70 from the right, C(w) ...
... a. The graph of same-day shipping function is discontinuous at each integral value of w in its domain because the function does not approach the same value from the left and the right. For example, as w approaches 70 from the left, C(w) approaches 229.25, but as w approaches 70 from the right, C(w) ...
Chapter 1 Introduction to Recursive Methods
... This is hence represented by a vertical line in the (k, c) space. Of course, only one level of consumption is consistent with feasibility. But feasibility is guaranteed by the function c0 . At the crossing point of the two curves one finds the pair (k ss , css ) which fully describes the steady stat ...
... This is hence represented by a vertical line in the (k, c) space. Of course, only one level of consumption is consistent with feasibility. But feasibility is guaranteed by the function c0 . At the crossing point of the two curves one finds the pair (k ss , css ) which fully describes the steady stat ...
Name - Bugbee
... A graph is concave up if f’ is increasing and concave down if f’ is decreasing. A graph is concave up if f” is + and concave down if f” is -. A point of inflection is a point at which the concavity changes. A function has a vertical asymptote at all x-values that make the denominator zero. f(x) = 1/ ...
... A graph is concave up if f’ is increasing and concave down if f’ is decreasing. A graph is concave up if f” is + and concave down if f” is -. A point of inflection is a point at which the concavity changes. A function has a vertical asymptote at all x-values that make the denominator zero. f(x) = 1/ ...
Chapter 2: Fundamentals of the Analysis of Algorithm Efficiency
... A way of comparing functions that ignores constant factors and small input sizes ...
... A way of comparing functions that ignores constant factors and small input sizes ...
What is a mathematical function?
... •The basic process of computation is fundamentally different in a Functional Programming Language than in an imperative language – Imperative language: operations are done and the results are stored in variables for later use. Management of variables is a constant concern and source of complexity fo ...
... •The basic process of computation is fundamentally different in a Functional Programming Language than in an imperative language – Imperative language: operations are done and the results are stored in variables for later use. Management of variables is a constant concern and source of complexity fo ...
Partial Evaluation
... INPUTstatic: the part of the input data known at compile time INPUTdynamic: the part of the input data known at run time If we write it like this: P: INPUTstatic → {INPUTdynamic → OUTPUT} The residual in the bracket can be seen as specialized program. A partial evaluator is an algorithm which, when ...
... INPUTstatic: the part of the input data known at compile time INPUTdynamic: the part of the input data known at run time If we write it like this: P: INPUTstatic → {INPUTdynamic → OUTPUT} The residual in the bracket can be seen as specialized program. A partial evaluator is an algorithm which, when ...
Decorators in Python
... before and after a function is called. A decorator takes a function object as an argument (which is called the decoratee) and returns a new function object that will be executed in its place. The function object constructed inside the decorator usually calls the decoratee. A decorator is executed ...
... before and after a function is called. A decorator takes a function object as an argument (which is called the decoratee) and returns a new function object that will be executed in its place. The function object constructed inside the decorator usually calls the decoratee. A decorator is executed ...
View/Open
... [G] graph twoway function gives several examples of how function graphs may be drawn on the fly. The manual entry does not quite explain the full flexibility and versatility of the command. Here is a further advertisement on its behalf. To underline a key feature: you do not need to create variables ...
... [G] graph twoway function gives several examples of how function graphs may be drawn on the fly. The manual entry does not quite explain the full flexibility and versatility of the command. Here is a further advertisement on its behalf. To underline a key feature: you do not need to create variables ...
g(n)
... • Definition. f(n) has the higher order growth then g(n), if the ratio f(n)/g(n) goes to infinity as n goes to infinity. • Note: both f(n) and g(n) are functions taking on positive values and they are increasing functions starting from a certain point. ...
... • Definition. f(n) has the higher order growth then g(n), if the ratio f(n)/g(n) goes to infinity as n goes to infinity. • Note: both f(n) and g(n) are functions taking on positive values and they are increasing functions starting from a certain point. ...
Recursive Functions of Symbolic Expressions and Their Application
... If f is an S-expression for an S-function f 0 and args is a list of arguments of the form (arg1 , . . . , argn ) where arg1 , . . . , argn are S-expressions, Then apply[f ; args] and f 0 [arg1 , . . . , argn ] are defined for the same values of arg1 , . . . , argn and are equal when defined. example ...
... If f is an S-expression for an S-function f 0 and args is a list of arguments of the form (arg1 , . . . , argn ) where arg1 , . . . , argn are S-expressions, Then apply[f ; args] and f 0 [arg1 , . . . , argn ] are defined for the same values of arg1 , . . . , argn and are equal when defined. example ...
Foundations of Functional Programming
... This course is concerned with the -calculus and its close relative, combinatory logic. The -calculus is important to functional programming and to computer science generally: 1. Variable binding and scoping in block-structured languages can be modelled. 2. Several function calling mechanisms — call- ...
... This course is concerned with the -calculus and its close relative, combinatory logic. The -calculus is important to functional programming and to computer science generally: 1. Variable binding and scoping in block-structured languages can be modelled. 2. Several function calling mechanisms — call- ...
Functional Programming: Introduction Introduction (Cont.)
... – Semantics were redefined in Common LISP. – Common LISP is now the standard version of LISP. • Designed by committee in 1980s. – Simple syntax, but a large language. • Scheme – Created in 1975 by Gerald Sussman and Guy Steele. – A dialect of LISP – Simple syntax, small language – Close to the initi ...
... – Semantics were redefined in Common LISP. – Common LISP is now the standard version of LISP. • Designed by committee in 1980s. – Simple syntax, but a large language. • Scheme – Created in 1975 by Gerald Sussman and Guy Steele. – A dialect of LISP – Simple syntax, small language – Close to the initi ...
Algorithms
... algorithm (using same or different programming languages) Given same input, different implementations of an algorithm should produce the same output For example, an algorithm is like a story, and an implementation is like a book on the story ...
... algorithm (using same or different programming languages) Given same input, different implementations of an algorithm should produce the same output For example, an algorithm is like a story, and an implementation is like a book on the story ...
Decrease-and
... Two nested loops each runs up to N. The body contains a function with complexity O(logN). Hence the complexity of probRelPrime is O(N2logN) 5. Conclusion The Decrease-and-Conquer paradigm relies on a relation between an instance of the problem and a smaller instance of the same problem. It may happe ...
... Two nested loops each runs up to N. The body contains a function with complexity O(logN). Hence the complexity of probRelPrime is O(N2logN) 5. Conclusion The Decrease-and-Conquer paradigm relies on a relation between an instance of the problem and a smaller instance of the same problem. It may happe ...
Foundations of Programming Languages * Course Overview
... – No loops. Use recursion instead. Why? ...
... – No loops. Use recursion instead. Why? ...
Document
... concat, a function that concatenates two lists. 6. Using the primitives and functionals, define IG, the index generator function. (IG n) creates a list of length n containing the first n positive integers. E.g. (IG 5) returns (1 2 3 4 5) ...
... concat, a function that concatenates two lists. 6. Using the primitives and functionals, define IG, the index generator function. (IG n) creates a list of length n containing the first n positive integers. E.g. (IG 5) returns (1 2 3 4 5) ...
PDF
... the type Eq : (D × D) → P rop. Thus for two elements of D, we can form the proposition Eq(x, y). Intuitively we understand what equality means on numbers, but we will see that this relation is subtle, and the axiomatic approach is abstract, so we don’t have a strong concrete idea of equality when we ...
... the type Eq : (D × D) → P rop. Thus for two elements of D, we can form the proposition Eq(x, y). Intuitively we understand what equality means on numbers, but we will see that this relation is subtle, and the axiomatic approach is abstract, so we don’t have a strong concrete idea of equality when we ...
Module 4
... • Sometimes, the best way to solve a problem is by solving a smaller version of the exact same problem first. • Recursion is a technique that solves a problem by solving a smaller problem of the same type. • A function that calls itself again and again is called as recursive function. • Advantage is ...
... • Sometimes, the best way to solve a problem is by solving a smaller version of the exact same problem first. • Recursion is a technique that solves a problem by solving a smaller problem of the same type. • A function that calls itself again and again is called as recursive function. • Advantage is ...
fundamentals of algorithms
... programming algorithm generally involves two separate steps: • Formulate problem recursively. Write down a formula for the whole problem as a simple combination of answers to smaller sub-problems. • Build solution to recurrence from bottom up. Write an algorithm that starts with base cases and works ...
... programming algorithm generally involves two separate steps: • Formulate problem recursively. Write down a formula for the whole problem as a simple combination of answers to smaller sub-problems. • Build solution to recurrence from bottom up. Write an algorithm that starts with base cases and works ...
Mathematical induction - Department of Information Technology
... However, the natural numbers is far from the only structure which can be defined as such: sky's the limit! Linked lists and binary trees are examples of well-known data structures that can be inductively defined (the definitions are left as an exercise for the reader). Clearly, if other structures c ...
... However, the natural numbers is far from the only structure which can be defined as such: sky's the limit! Linked lists and binary trees are examples of well-known data structures that can be inductively defined (the definitions are left as an exercise for the reader). Clearly, if other structures c ...
Universal language Decision problems Reductions Post`s
... Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: to devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. Theorem (Matiyasevich 1970) Hilbert’ ...
... Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: to devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. Theorem (Matiyasevich 1970) Hilbert’ ...
Limit Definition of the Derivative
... There is a theorem relating differentiability and continuity which says that if a function is differentiable at a point then it is also continuous there. The contrapositive of this theorem is that if a function is not continuous at a point then it is not differentiable there, so this gives us one wa ...
... There is a theorem relating differentiability and continuity which says that if a function is differentiable at a point then it is also continuous there. The contrapositive of this theorem is that if a function is not continuous at a point then it is not differentiable there, so this gives us one wa ...
Recursion (computer science)
Recursion in computer science is a method where the solution to a problem depends on solutions to smaller instances of the same problem (as opposed to iteration). The approach can be applied to many types of problems, and recursion is one of the central ideas of computer science.""The power of recursion evidently lies in the possibility of defining an infinite set of objects by a finite statement. In the same manner, an infinite number of computations can be described by a finite recursive program, even if this program contains no explicit repetitions.""Most computer programming languages support recursion by allowing a function to call itself within the program text. Some functional programming languages do not define any looping constructs but rely solely on recursion to repeatedly call code. Computability theory proves that these recursive-only languages are Turing complete; they are as computationally powerful as Turing complete imperative languages, meaning they can solve the same kinds of problems as imperative languages even without iterative control structures such as “while” and “for”.